Polynomials ABC

Polynomials abcFrom Wikipedia, the free encyclopedia

Contents

1 Abel polynomials 11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 AbelRuni theorem 32.1 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Lower-degree polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Quintics and higher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Actuarial polynomials 73.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Additive polynomial 84.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 The ring of additive polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 The fundamental theorem of additive polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 94.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Alexander polynomial 105.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Computing the polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.3 Basic properties of the polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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5.4 Geometric signicance of the polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.5 Relations to satellite operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.6 AlexanderConway polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.7 Relation to Khovanov homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Algebraic equation 146.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 Areas of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7 Algebraic variety 177.1 Introduction and denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7.1.1 Ane varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.1.2 Projective varieties and quasi-projective varieties . . . . . . . . . . . . . . . . . . . . . . . 197.1.3 Abstract varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2.1 Subvariety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.2.2 Ane variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.2.3 Projective variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7.3 Basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.4 Isomorphism of algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.5 Discussion and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.6 Algebraic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.8 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

8 All one polynomial 258.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

9 Almost linear hash function 279.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

10 Alternating polynomial 2910.1 Relation to symmetric polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.2 Vandermonde polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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10.2.1 Ring structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.3 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.4 Unstable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

11 Angelescu polynomials 3211.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

12 Appell sequence 3312.1 Equivalent characterizations of Appell sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.2 Recursion formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.3 Subgroup of the Sheer polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.4 Dierent convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

13 Bell polynomials 3713.1 Complete Bell polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713.2 Combinatorial meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

13.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

13.3.1 Stirling numbers and Bell numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.3.2 Touchard polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.3.3 Convolution identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

13.4 Applications of Bell polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.4.1 Fa di Brunos formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.4.2 Moments and cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.4.3 Representation of polynomial sequences of binomial type . . . . . . . . . . . . . . . . . . 40

13.5 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4113.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4113.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

14 Bernoulli polynomials 4314.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

14.1.1 Explicit formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.1.2 Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.1.3 Representation by a dierential operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.1.4 Representation by an integral operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

14.2 Another explicit formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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14.3 Sums of pth powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4614.4 The Bernoulli and Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4614.5 Explicit expressions for low degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4614.6 Maximum and minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.7 Dierences and derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

14.7.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.7.2 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

14.8 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.9 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.10Relation to falling factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.11Multiplication theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.12Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.13Periodic Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

15 Bernstein polynomial 5315.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.4 Approximating continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

15.4.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5715.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5715.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

16 BernsteinSato polynomial 5816.1 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

17 Binomial 6117.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.2 Operations on simple binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6217.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6217.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

18 BoasBuck polynomials 6318.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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19 BollobsRiordan polynomial 6419.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

20 Bombieri norm 6620.1 Bombieri scalar product for homogeneous polynomials . . . . . . . . . . . . . . . . . . . . . . . . 6620.2 Bombieri inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.3 Invariance by isometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.4 Other inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

21 Boole polynomials 6921.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6921.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

22 Bracket polynomial 7022.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

23 Bring radical 7123.1 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

23.1.1 Principal quintic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.1.2 BringJerrard normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7223.1.3 Brioschi normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

23.2 Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7323.3 Solution of the general quintic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7323.4 Other characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

23.4.1 The HermiteKroneckerBrioschi characterization . . . . . . . . . . . . . . . . . . . . . . 7423.4.2 Glassers derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7623.4.3 The method of dierential resolvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7823.4.4 DoyleMcMullen iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

23.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8023.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8023.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8123.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

24 Bzout matrix 8224.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8224.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

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24.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8324.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8324.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

25 Caloric polynomial 8525.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8525.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

26 Casus irreducibilis 8626.1 Formal statement and proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8626.2 Solution in non-real radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.3 Non-algebraic solution in terms of real quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.4 Relation to angle trisection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.5 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8826.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8826.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8826.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

27 Cavalieris quadrature formula 8927.1 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

27.1.1 Negative n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9027.1.2 n = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9027.1.3 Alternative forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

27.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9227.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

27.4.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9327.4.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

27.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

28 Characteristic polynomial 9628.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.5 Characteristic polynomial of a product of two matrices . . . . . . . . . . . . . . . . . . . . . . . . 9828.6 Secular function and secular equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

28.6.1 Secular function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9928.6.2 Secular equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

28.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9928.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9928.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

CONTENTS vii

29 Coecient 10029.1 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10029.2 Examples of physical coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10129.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10129.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

30 Coecient diagram method 10230.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10330.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10330.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

31 Complex conjugate root theorem 10431.1 Examples and consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

31.1.1 Corollary on odd-degree polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10431.2 Simple proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10531.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

32 Complex quadratic polynomial 10732.1 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10732.2 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

32.2.1 Between forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10732.2.2 With doubling map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

32.3 Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10832.4 Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10832.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10832.6 Critical items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

32.6.1 Critical point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10932.6.2 Critical value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10932.6.3 Critical orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10932.6.4 Critical sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11032.6.5 Critical polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11032.6.6 Critical curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

32.7 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11132.7.1 Parameter plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11232.7.2 Dynamical plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

32.8 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.8.1 Derivative with respect to c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.8.2 Derivative with respect to z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11532.8.3 Schwarzian derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

32.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11632.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11632.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

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33 Constant function 11833.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11833.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11833.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12033.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

34 Constant term 12134.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

35 Content (algebra) 12335.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12335.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

36 Continuant (mathematics) 12436.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12436.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12436.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

37 Cubic function 12637.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12737.2 Critical points of a cubic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13037.3 Roots of a cubic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

37.3.1 The nature of the roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13037.3.2 General formula for roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13237.3.3 Reduction to a depressed cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13337.3.4 Cardanos method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13337.3.5 Vietas substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13537.3.6 Lagranges method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13637.3.7 Trigonometric (and hyperbolic) method . . . . . . . . . . . . . . . . . . . . . . . . . . . 13737.3.8 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13837.3.9 Geometric interpretation of the roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

37.4 Collinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14037.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14037.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14037.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14037.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14237.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

38 Cyclotomic polynomial 14738.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14738.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

38.2.1 Fundamental tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14938.2.2 Easy cases for the computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

CONTENTS ix

38.2.3 Integers appearing as coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15038.2.4 Gausss formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15038.2.5 Lucass formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

38.3 Cyclotomic polynomials over Zp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15138.4 Prime Cyclotomic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15238.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15238.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15238.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15238.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15338.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

39 List of polynomial topics 15439.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15439.2 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

39.2.1 Elementary abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15539.3 Theory of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15539.4 Calculus with polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15639.5 Polynomial interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15639.6 Weierstrass approximation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15639.7 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15639.8 Named polynomials and polynomial sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15739.9 Knot polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15839.10Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15839.11Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 159

39.11.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15939.11.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16239.11.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Chapter 1

Abel polynomials

The Abel polynomials in mathematics form a polynomial sequence, the nth term of which is of the form

pn(x) = x(x an)n1:The sequence is named after Niels Henrik Abel (1802-1829), the Norwegian mathematician.This polynomial sequence is of binomial type: conversely, every polynomial sequence of binomial type may beobtained from the Abel sequence in the umbral calculus.

1.1 ExamplesFor a=1, the polynomials are (sequence A137452 in OEIS)

p0(x) = 1;

p1(x) = x;

p2(x) = 2x+ x2;p3(x) = 9x 6x2 + x3;p4(x) = 64x+ 48x2 12x3 + x4;For a=2, the polynomials are

p0(x) = 1;

p1(x) = x;

p2(x) = 4x+ x2;p3(x) = 36x 12x2 + x3;p4(x) = 512x+ 192x2 24x3 + x4;p5(x) = 10000x 4000x2 + 600x3 40x4 + x5;p6(x) = 248832x+ 103680x2 17280x3 + 1440x4 60x5 + x6;

1.2 References Rota, Gian-Carlo; Shen, Jianhong; Taylor, Brian D. (1997). All Polynomials of Binomial Type Are Repre-sented by Abel Polynomials. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze Sr. 4 25 (34):731738. MR 1655539. Zbl 1003.05011.

1

2 CHAPTER 1. ABEL POLYNOMIALS

1.3 External links Weisstein, Eric W., Abel Polynomial, MathWorld.

Chapter 2

AbelRuni theorem

Not to be confused with Abels theorem.

x = bpb24ac2a

A general solution to any quadratic equation can be given using the quadratic formula above. Similar formulas existfor polynomial equations of degree 3 and 4. But no such formula is possible for 5th degree polynomials; the realsolution 1.1673... to the 5th degree equation below cannot be written using basic arithmetic operations and nthroots:x5 x+ 1 = 0In algebra, the AbelRuni theorem (also known as Abels impossibility theorem) states that there is no gen-eral algebraic solutionthat is, solution in radicals to polynomial equations of degree ve or higher with arbitrarycoecients.[1] The theorem is named after Paolo Runi, who made an incomplete proof in 1799, and Niels HenrikAbel, who provided a proof in 1823. variste Galois independently proved the theorem in a work that was posthu-mously published in 1846.[2]

2.1 InterpretationThe theorem does not assert that some higher-degree polynomial equations have no solution. In fact, the oppositeis true: every non-constant polynomial equation in one unknown, with real or complex coecients, has at least onecomplex number as a solution (and thus, by polynomial division, as many complex roots as its degree, countingrepeated roots); this is the fundamental theorem of algebra. These solutions can be computed to any desired degreeof accuracy using numerical methods such as the NewtonRaphson method or Laguerre method, and in this way theyare no dierent from solutions to polynomial equations of the second, third, or fourth degrees. The theorem onlyshows that the solutions of some of these equations cannot be expressed via a general expression in radicals.Also, the theorem does not assert that some higher-degree polynomial equations have roots which cannot be expressedin terms of radicals. While this is now known to be true, it is a stronger claim, which was only proven a few yearslater by Galois. The theorem only shows that there is no general solution in terms of radicals which gives the roots toa generic polynomial with arbitrary coecients. It did not by itself rule out the possibility that each polynomial maybe solved in terms of radicals on a case-by-case basis.

2.2 Lower-degree polynomialsThe solutions of any second-degree polynomial equation can be expressed in terms of addition, subtraction, multi-plication, division, and square roots, using the familiar quadratic formula: The roots of the following equation areshown below:

ax2 + bx+ c = 0; a 6= 0

3

4 CHAPTER 2. ABELRUFFINI THEOREM

x =bpb2 4ac

2a:

Analogous formulas for third- and fourth-degree equations, using cube roots and fourth roots, have been known sincethe 16th century.

2.3 Quintics and higherThe AbelRuni theorem says that there are some fth-degree equations whose solution cannot be so expressed.The equation x5 x + 1 = 0 is an example. (See Bring radical.) Some other fth degree equations can be solvedby radicals, for example x5 x4 x + 1 = 0 , which factors into (x 1)(x 1)(x + 1)(x + i)(x i) = 0 .The precise criterion that distinguishes between those equations that can be solved by radicals and those that cannotwas given by variste Galois and is now part of Galois theory: a polynomial equation can be solved by radicals if andonly if its Galois group (over the rational numbers, or more generally over the base eld of admitted constants) is asolvable group.Today, in the modern algebraic context, we say that second, third and fourth degree polynomial equations can alwaysbe solved by radicals because the symmetric groups S2, S3 and S4 are solvable groups, whereas Sn is not solvablefor n 5. This is so because for a polynomial of degree n with indeterminate coecients (i.e., given by symbolicparameters), the Galois group is the full symmetric group Sn (this is what is called the general equation of the n-thdegree). This remains true if the coecients are concrete but algebraically independent values over the base eld.

2.4 ProofThe following proof is based on Galois theory (for a short explanation of Arnolds proof that does not rely on priorknowledge in group theory see [3]). Historically, Runi and Abels proofs precede Galois theory and Arnolds. Fora modern presentation of Abels proof see the books of Tignol or Pesic.One of the fundamental theorems of Galois theory states that a polynomial f(x) F[x] is solvable by radicals overF if and only if its splitting eld K over F has a solvable Galois group,[4] so the proof of the AbelRuni theoremcomes down to computing the Galois group of the general polynomial of the fth degree.Let y1 be a real number transcendental over the eld of rational numbersQ , and let y2 be a real number transcendentalover Q(y1) , and so on to y5 which is transcendental over Q(y1; y2; y3; y4) . These numbers are called independenttranscendental elements over Q. Let E = Q(y1; y2; y3; y4; y5) and let

f(x) = (x y1)(x y2)(x y3)(x y4)(x y5) 2 E[x]:Expanding f(x) out yields the elementary symmetric functions of the yn :

s1 = y1 + y2 + y3 + y4 + y5

s2 = y1y2 + y1y3 + y1y4 + y1y5 + y2y3 + y2y4 + y2y5 + y3y4 + y3y5 + y4y5

s3 = y1y2y3 + y1y2y4 + y1y2y5 + y1y3y4 + y1y3y5 + y1y4y5 + y2y3y4 + y2y3y5 + y2y4y5 + y3y4y5

s4 = y1y2y3y4 + y1y2y3y5 + y1y2y4y5 + y1y3y4y5 + y2y3y4y5

s5 = y1y2y3y4y5:

The coecient of xn in f(x) is thus (1)5ns5n . Let F = Q(si) be the eld obtained by adjoining the symmetricfunctions to the rationals (the si are all transcendental, because the yi are independent). Because our independenttranscendentals yn act as indeterminates overQ , every permutation in the symmetric group on 5 letters S5 inducesa distinct automorphism 0 onE that leavesQ xed and permutes the elements yn . Since an arbitrary rearrangementof the roots of the product form still produces the same polynomial, e.g.:

(y y3)(y y1)(y y2)(y y5)(y y4)

2.5. HISTORY 5

is still the same polynomial as

(y y1)(y y2)(y y3)(y y4)(y y5)

the automorphisms 0 also leave f xed, so they are elements of the Galois group G(E/F ) . So we have shownthat S5 G(E/F ) ; however there could possibly be automorphisms there that are not in S5 . However, since therelative automorphism group for the splitting eld of a quintic polynomial has at most 5! elements, it follows thatG(E/F ) is isomorphic to S5 . Generalizing this argument shows that the Galois group of every general polynomialof degree n is isomorphic to Sn .And what of S5 ? The only composition series of S5 is S5 A5 feg (where A5 is the alternating group on veletters, also known as the icosahedral group). However, the quotient groupA5/feg (isomorphic toA5 itself) is not anabelian group, and so S5 is not solvable, so it must be that the general polynomial of the fth degree has no solutionin radicals. Since the rst nontrivial normal subgroup of the symmetric group on n letters is always the alternatinggroup on n letters, and since the alternating groups on n letters for n 5 are always simple and non-abelian, andhence not solvable, it also says that the general polynomials of all degrees higher than the fth also have no solutionin radicals.Note that the above construction of the Galois group for a fth degree polynomial only applies to the general poly-nomial, specic polynomials of the fth degree may have dierent Galois groups with quite dierent properties, e.g.x5 1 has a splitting eld generated by a primitive 5th root of unity, and hence its Galois group is abelian and theequation itself solvable by radicals; moreover the argument does not provide any rational-valued quintic that has S5or A5 as its Galois group. However, since the result is on the general polynomial, it does say that a general quinticformula for the roots of a quintic using only a nite combination of the arithmetic operations and radicals in termsof the coecients is impossible. Q.E.D.

2.5 HistoryAround 1770, Joseph Louis Lagrange began the groundwork that unied the many dierent tricks that had been usedup to that point to solve equations, relating them to the theory of groups of permutations, in the form of Lagrangeresolvents. This innovative work by Lagrange was a precursor to Galois theory, and its failure to develop solutions forequations of fth and higher degrees hinted that such solutions might be impossible, but it did not provide conclusiveproof. The theorem, however, was rst nearly proved by Paolo Runi in 1799, but his proof was mostly ignored.He had several times tried to send it to dierent mathematicians to get it acknowledged, amongst them, Frenchmathematician Augustin-Louis Cauchy, but it was never acknowledged, possibly because the proof was spanning 500pages. The proof also, as was discovered later, contained an error. In modern terms, Runi failed to prove that thesplitting eld is one of the elds in the tower of radicals which corresponds to the hypothesized solution by radicals;this assumption fails, for example, for Cardanos solution of the cubic; it splits not only the original cubic but alsothe two others with the same discriminant. While Cauchy felt that the assumption was minor, most historians believethat the proof was not complete until Abel proved this assumption. The theorem is thus generally credited to NielsHenrik Abel, who published a proof that required just six pages in 1824.[5]

Proving that some quintic (and higher) equations were unsolvable by radicals did not completely settle the matter,because the AbelRuni theorem does not provide necessary and sucient conditions for saying precisely whichquintic (and higher) equations are unsolvable by radicals; for example x5 1 = 0 is solvable. Abel was working on acomplete characterization when he died in 1829.[6] Furthermore, Ian Stewart notes that for all that Abels methodscould prove, every particular quintic equation might be soluble, with a special formula for each equation.[7]

In 1830 Galois (at the age of 18) submitted to the Paris Academy of Sciences a memoir on his theory of solvabilityby radicals; Galois paper was ultimately rejected in 1831 as being too sketchy and for giving a condition in termsof the roots of the equation instead of its coecients. Galois then died in 1832 and his paper"Memoire sur lesconditions de resolubilite des equations par radicauxremained unpublished until 1846 when it was published byJoseph Liouville accompanied by some of his own explanations.[6] Prior to this publication, Liouville announcedGalois result to theAcademy in a speech he gave on 4 July 1843.[8] According toAllan Clark, Galoiss characterizationdramatically supersedes the work of Abel and Runi.[9]

In 1963, Vladimir Arnold discovered a topological proof of the AbelRuni theorem,[10] which served as a startingpoint for topological Galois theory.[11]

6 CHAPTER 2. ABELRUFFINI THEOREM

2.6 See also Theory of equations Constructible number

2.7 Notes[1] Jacobson (2009), p. 211.[2] Galois, variste (1846). OEuvres mathmatiques d'variste Galois.. Journal des mathmatiques pures et appliques XI:

381444. Retrieved 2009-02-04.[3] Short proof of Abels theorem that 5th degree polynomial equations cannot be solved.[4] Fraleigh (1994, p. 401)[5] du Sautoy, Marcus. January: Impossibilities. Symmetry: A Journey into the Patterns of Nature. ISBN 978-0-06-078941-

1.[6] Jean-Pierre Tignol (2001). Galois Theory of Algebraic Equations. World Scientic. pp. 232233 and 302. ISBN 978-

981-02-4541-2.[7] Stewart, 3rd ed., p. xxiii[8] Stewart, 3rd ed., p. xxiii[9] Allan Clark (1984) [1971]. Elements of Abstract Algebra. Courier Corporation. p. 131. ISBN 978-0-486-14035-3.[10] Tribute toVladimir Arnold (PDF).Notices of the AmericanMathematical Society 59 (3): 393. March 2012. doi:10.1090/noti810.[11] Vladimir Igorevich Arnold. 2010.

2.8 References Edgar Dehn. Algebraic Equations: An Introduction to the Theories of Lagrange and Galois. Columbia Univer-sity Press, 1930. ISBN 0-486-43900-3.

Jacobson, Nathan (2009), Basic algebra 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1 John B. Fraleigh. A First Course in Abstract Algebra. Fifth Edition. Addison-Wesley, 1994. ISBN 0-201-59291-6.

Ian Stewart. Galois Theory. Chapman and Hall, 1973. ISBN 0-412-10800-3. Abels Impossibility Theorem at Everything2

2.9 Further reading Peter Pesic (2003). Abels Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability. MITPress. ISBN 978-0-262-66182-9.

2.10 External links Mmoire sur les quations algbriques, ou l'on dmontre l'impossibilit de la rsolution de l'quation gnraledu cinquime degr PDF - the rst proof in French (1824)

Dmonstration de l'impossibilit de la rsolution algbrique des quations gnrales qui passent le quatrimedegr PDF - the second proof in French (1826)

A video presentation on Arnolds proof.

Chapter 3

Actuarial polynomials

In mathematics, the actuarial polynomials a()n(x) are polynomials studied by Toscano (1950) given by the generating function

Xn

a()n (x)

n!tn = exp(t+ x(1 et))

(Roman 1984, 4.3.4), Boas & Buck (1958).

3.1 See also Umbral calculus

3.2 References Boas, Ralph P.; Buck, R. Creighton (1958), Polynomial expansions of analytic functions, Ergebnisse der Math-ematik und ihrer Grenzgebiete. Neue Folge. 19, Berlin, New York: Springer-Verlag, MR 0094466

Roman, Steven (1984), The umbral calculus, Pure and Applied Mathematics 111, London: Academic PressInc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-594380-2, MR 741185 Reprinted by Dover,2005

Toscano, Letterio (1950), Una classe di polinomi della matematica attuariale, Rivista di Matematica dellaUniversit di Parma (in Italian) 1: 459470, MR 0040480, Zbl 0040.03204

7

Chapter 4

Additive polynomial

In mathematics, the additive polynomials are an important topic in classical algebraic number theory.

4.1 DenitionLet k be a eld of characteristic p, with p a prime number. A polynomial P(x) with coecients in k is called anadditive polynomial, or a Frobenius polynomial, if

P (a+ b) = P (a) + P (b)

as polynomials in a and b. It is equivalent to assume that this equality holds for all a and b in some innite eldcontaining k, such as its algebraic closure.Occasionally absolutely additive is used for the condition above, and additive is used for the weaker condition thatP(a + b) = P(a) + P(b) for all a and b in the eld. For innite elds the conditions are equivalent, but for nite eldsthey are not, and the weaker condition is the wrong one and does not behave well. For example, over a eld oforder q any multiple P of xq x will satisfy P(a + b) = P(a) + P(b) for all a and b in the eld, but will usually not be(absolutely) additive.

4.2 ExamplesThe polynomial xp is additive. Indeed, for any a and b in the algebraic closure of k one has by the binomial theorem

(a+ b)p =

pXn=0

p

n

anbpn:

Since p is prime, for all n = 1, ..., p1 the binomial coecient (pn) is divisible by p, which implies that

(a+ b)p ap + bp mod p

as polynomials in a and b.Similarly all the polynomials of the form

np (x) = xpn

are additive, where n is a non-negative integer.

8

4.3. THE RING OF ADDITIVE POLYNOMIALS 9

4.3 The ring of additive polynomialsIt is quite easy to prove that any linear combination of polynomials np (x) with coecients in k is also an additivepolynomial. An interesting question is whether there are other additive polynomials except these linear combinations.The answer is that these are the only ones.One can check that if P(x) and M(x) are additive polynomials, then so are P(x) + M(x) and P(M(x)). These implythat the additive polynomials form a ring under polynomial addition and composition. This ring is denoted

kfpg:

This ring is not commutative unless k equals the eld Fp=Z/pZ (see modular arithmetic). Indeed, consider the additivepolynomials ax and xp for a coecient a in k. For them to commute under composition, we must have

(ax)p = axp;

or ap a = 0. This is false for a not a root of this equation, that is, for a outside Fp:

4.4 The fundamental theorem of additive polynomialsLet P(x) be a polynomial with coecients in k, and fw1;:::;wmgk be the set of its roots. Assuming that the roots ofP(x) are distinct (that is, P(x) is separable), then P(x) is additive if and only if the set fw1;:::;wmg forms a group withthe eld addition.

4.5 See also Drinfeld module Additive map

4.6 References David Goss, Basic Structures of Function Field Arithmetic, 1996, Springer, Berlin. ISBN 3-540-61087-1.

4.7 External links Weisstein, Eric W., Additive Polynomial, MathWorld.

Chapter 5

Alexander polynomial

In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coecientsto each knot type. James Waddell Alexander II discovered this, the rst knot polynomial, in 1923. In 1969, JohnConway showed a version of this polynomial, now called the AlexanderConway polynomial, could be computedusing a skein relation, although its signicance was not realized until the discovery of the Jones polynomial in 1984.Soon after Conways reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibitedin Alexanders paper on his polynomial.[1]

5.1 DenitionLet K be a knot in the 3-sphere. Let X be the innite cyclic cover of the knot complement of K. This covering can beobtained by cutting the knot complement along a Seifert surface of K and gluing together innitely many copies ofthe resulting manifold with boundary in a cyclic manner. There is a covering transformation t acting on X. Considerthe rst homology (with integer coecients) of X, denotedH1(X) . The transformation t acts on the homology andso we can considerH1(X) a module over Z[t; t1] . This is called the Alexander invariant or Alexander module.The module is nitely presentable; a presentation matrix for this module is called the Alexander matrix. If thenumber of generators, r, is less than or equal to the number of relations, s, then we consider the ideal generated byall r by r minors of the matrix; this is the zero'th Fitting ideal or Alexander ideal and does not depend on choiceof presentation matrix. If r > s, set the ideal equal to 0. If the Alexander ideal is principal, take a generator; this iscalled an Alexander polynomial of the knot. Since this is only unique up to multiplication by the Laurent monomialtn , one often xes a particular unique form. Alexanders choice of normalization is to make the polynomial havea positive constant term.Alexander proved that the Alexander ideal is nonzero and always principal. Thus an Alexander polynomial alwaysexists, and is clearly a knot invariant, denoted K(t) . The Alexander polynomial for the knot congured by onlyone string is a polynomial of t2 and then it is the same polynomial for the mirror image knot. Namely, it can notdistinguish between the knot and one for its mirror image.

5.2 Computing the polynomialThe following procedure for computing the Alexander polynomial was given by J. W. Alexander in his paper.Take an oriented diagram of the knot with n crossings; there are n + 2 regions of the knot diagram. To work out theAlexander polynomial, rst one must create an incidence matrix of size (n, n + 2). The n rows correspond to the ncrossings, and the n + 2 columns to the regions. The values for the matrix entries are either 0, 1, 1, t, t.Consider the entry corresponding to a particular region and crossing. If the region is not adjacent to the crossing, theentry is 0. If the region is adjacent to the crossing, the entry depends on its location. The following table gives theentry, determined by the location of the region at the crossing from the perspective of the incoming undercrossingline.

on the left before undercrossing: t

10

5.3. BASIC PROPERTIES OF THE POLYNOMIAL 11

on the right before undercrossing: 1on the left after undercrossing: ton the right after undercrossing: 1

Remove two columns corresponding to adjacent regions from the matrix, and work out the determinant of the newn by n matrix. Depending on the columns removed, the answer will dier by multiplication by tn . To resolve thisambiguity, divide out the largest possible power of t and multiply by 1 if necessary, so that the constant term ispositive. This gives the Alexander polynomial.The Alexander polynomial can also be computed from the Seifert matrix.After the work of Alexander R. Fox considered a copresentation of the knot group 1(S3nK) , and introduced non-commutative dierential calculus Fox (1961), which also permits one to compute K(t) . Detailed exposition ofthis approach about higher Alexander polynomials can be found in the book Crowell & Fox (1963).

5.3 Basic properties of the polynomialThe Alexander polynomial is symmetric: K(t1) = K(t) for all knots K.

From the point of view of the denition, this is an expression of the Poincar Duality isomorphismH1X ' HomZ[t;t1](H1X;G) whereG is the quotient of the eld of fractions of Z[t; t1] by Z[t; t1], considered as a Z[t; t1] -module, and where H1X is the conjugate Z[t; t1] -module to H1X ie: asan abelian group it is identical to H1X but the covering transformation t acts by t1 .

and it evaluates to a unit on 1: K(1) = 1 .

From the point of view of the denition, this is an expression of the fact that the knot complement is ahomology circle, generated by the covering transformation t . More generally ifM is a 3-manifold suchthat rank(H1M) = 1 it has an Alexander polynomial M (t) dened as the order ideal of its innite-cyclic covering space. In this case M (1) is, up to sign, equal to the order of the torsion subgroup ofH1M .

It is known that every integral Laurent polynomial which is both symmetric and evaluates to a unit at 1 is the Alexanderpolynomial of a knot (Kawauchi 1996).

5.4 Geometric signicance of the polynomialSince the Alexander ideal is principal,K(t) = 1 if and only if the commutator subgroup of the knot group is perfect(i.e. equal to its own commutator subgroup).For a topologically slice knot, the Alexander polynomial satises the FoxMilnor condition K(t) = f(t)f(t1)where f(t) is some other integral Laurent polynomial.Twice the knot genus is bounded below by the degree of the Alexander polynomial.Michael Freedman proved that a knot in the 3-sphere is topologically slice; i.e., bounds a locally-at topologicaldisc in the 4-ball, if the Alexander polynomial of the knot is trivial (Freedman and Quinn, 1990).Kauman (1983) describes the rst construction of the Alexander polynomial via state sums derived from physicalmodels. A survey of these topic and other connections with physics are given in Kauman (2001).There are other relations with surfaces and smooth 4-dimensional topology. For example, under certain assumptions,there is a way of modifying a smooth 4-manifold by performing a surgery that consists of removing a neighborhood ofa two-dimensional torus and replacing it with a knot complement crossed with S1. The result is a smooth 4-manifoldhomeomorphic to the original, though now the SeibergWitten invariant has been modied by multiplication withthe Alexander polynomial of the knot.[2]

Knots with symmetries are known to have restricted Alexander polynomials. See the symmetry section in (Kawauchi1996). Nonetheless, the Alexander polynomial can fail to detect some symmetries, such as strong invertibility.

12 CHAPTER 5. ALEXANDER POLYNOMIAL

If the knot complement bers over the circle, then the Alexander polynomial of the knot is known to be monic (thecoecients of the highest and lowest order terms are equal to 1 ). In fact, if S ! CK ! S1 is a ber bundlewhere CK is the knot complement, let g : S ! S represent the monodromy, then K(t) = Det(tI g) whereg : H1S ! H1S is the induced map on homology.

5.5 Relations to satellite operationsIf a knot K is a satellite knot with companion K 0 i.e.: there exists an embedding f : S1 D2 ! S3 such thatK = f(K 0) where S1 D2 S3 is an unknotted solid torus, then K(t) = f(S1f0g)(ta)K0(t) . Wherea 2 Z is the integer that representsK 0 S1 D2 inH1(S1 D2) = Z .Examples: For a connect-sum K1#K2(t) = K1(t)K2(t) . If K is an untwisted Whitehead double, thenK(t) = 1 .

5.6 AlexanderConway polynomialAlexander proved the Alexander polynomial satises a skein relation. John Conway later rediscovered this in adierent form and showed that the skein relation together with a choice of value on the unknot was enough to deter-mine the polynomial. Conways version is a polynomial in z with integer coecients, denoted r(z) and called theAlexanderConway polynomial (also known as Conway polynomial or ConwayAlexander polynomial).Suppose we are given an oriented link diagram, where L+; L; L0 are link diagrams resulting from crossing andsmoothing changes on a local region of a specied crossing of the diagram, as indicated in the gure.

L+ L- L 0Here are Conways skein relations:

r(O) = 1 (where O is any diagram of the unknot)

r(L+)r(L) = zr(L0)

The relationship to the standardAlexander polynomial is given byL(t2) = rL(tt1) . HereLmust be properlynormalized (by multiplication of tn/2 ) to satisfy the skein relation (L+) (L) = (t1/2 t1/2)(L0) .Note that this relation gives a Laurent polynomial in t1/2.See knot theory for an example computing the Conway polynomial of the trefoil.

5.7. RELATION TO KHOVANOV HOMOLOGY 13

5.7 Relation to Khovanov homologyIn Ozsvath & Szabo (2004) and Rasmussen (2003) the Alexander polynomial is presented as Euler characteristic ofa complex, whose homology are isotopy invariants of the considered knot K , therefore Floer homology theory is acategorication of the Alexander polynomial. For detail, see Khovanov homology Khovanov (2003).

5.8 Notes[1] Alexander describes his skein relation toward the end of his paper under the heading miscellaneous theorems, which is

possibly why it got lost. Joan Birman mentions in her paper New points of view in knot theory (Bull. Amer. Math. Soc.(N.S.) 28 (1993), no. 2, 253287) that Mark Kidwell brought her attention to Alexanders relation in 1970.

[2] Fintushel and Stern (1997) Knots, links, and 4-manifolds

5.9 References Alexander, J. W. (1928). Topological invariants of knots and links. Trans. Amer. Math. Soc. 30 (2):275306. doi:10.2307/1989123.

Crowell, R.; Fox, R. (1963). Introduction to Knot Theory. Ginn and Co. after 1977 Springer Verlag. Adams, Colin C. (2004). The Knot Book: An elementary introduction to the mathematical theory of knots(Revised reprint of the 1994 original ed.). Providence, RI: American Mathematical Society. ISBN 0-8218-3678-1. (accessible introduction utilizing a skein relation approach)

Fox, R. (1961). A quick trip through knot theory, In Topology of ThreeManifold (Proceedings of 1961Topology Institute at Univ. of Georgia, edited by M.K.Fort ed.). Englewood Clis. N. J.: Prentice-Hall. pp.120167.

Freedman, Michael H.; Quinn, Frank (1990). Topology of 4-manifolds. Princeton Mathematical Series 39.Princeton, NJ: Princeton University Press. ISBN 0-691-08577-3.

Kauman, Louis (1983). Formal Knot Theory. Princeton University press. Kauman, Louis (2001). Knots and Physics. World Scientic Publishing Companey. Kawauchi, Akio (1996). A Survey of Knot Theory. Birkhauser. (covers several dierent approaches, explainsrelations between dierent versions of the Alexander polynomial)

Khovanov,M. (2006). Link homology and categorication. Proceedings of the ICM-2006. arXiv:math/0605339. Ozsvath, Peter; Szabo, Zoltan (2004). Holomorphic disks and knot invariants. Adv. Math., no., 58-6. Adv.Math. 186 (1): 58116. arXiv:math/0209056. Bibcode:2002math......9056O. doi:10.1016/j.aim.2003.05.001.class=math.GT

Rasmussen, J. (2003). Floer homology and knot complements. PhD thesis Harvard University. p. 6378.arXiv:math/0306378. Bibcode:2003math......6378R.

Rolfsen, Dale (1990). Knots and Links (2nd ed.). Berkeley, CA: Publish or Perish. ISBN 0-914098-16-0.(explains classical approach using the Alexander invariant; knot and link table with Alexander polynomials)

5.10 External links Hazewinkel, Michiel, ed. (2001), Alexander invariants, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

"Main Page" and "The Alexander-Conway Polynomial", The Knot Atlas. knot and link tables with computedAlexander and Conway polynomials

Chapter 6

Algebraic equation

In mathematics, an algebraic equation or polynomial equation is an equation of the form

P = Q

where P and Q are polynomials with coecients in some eld, often the eld of the rational numbers. For mostauthors, an algebraic equation is univariate, which means that it involves only one variable. On the other hand, apolynomial equation may involve several variables, in which case it is called multivariate and the term polynomialequation is usually preferred to algebraic equation.For example,

x5 3x+ 1 = 0is an algebraic equation with integer coecients and

y4 +xy

2=

x3

3 xy2 + y2 1

7

is a multivariate polynomial equation over the rationals.Some but not all polynomial equations with rational coecients have a solution that is an algebraic expression witha nite number of operations involving just those coecients (that is, can be solved algebraically). This can be donefor all such equations of degree one, two, three, or four; but for degree ve or more it can only be done for someequations but not for all. A large amount of research has been devoted to compute eciently accurate approximationsof the real or complex solutions of an univariate algebraic equation (see Root-nding algorithm) and of the commonsolutions of several multivariate polynomial equations (see System of polynomial equations).

6.1 HistoryThe study of algebraic equations is probably as old as mathematics: the Babylonian mathematicians, as early as 2000BC could solve some kind of quadratic equations (displayed on Old Babylonian clay tablets).The algebraic equations over the rationals with only one variable are also called univariate equations. They have avery long history. Ancient mathematicians wanted the solutions in the form of radical expressions, like x = 1+

p5

2for the positive solution of x2 x 1 = 0 . The ancient Egyptians knew how to solve equations of degree 2 inthis manner. In the 9th century Muhammad ibn Musa al-Khwarizmi and other Islamic mathematicians derived thequadratic formula, the general solution of equations of degree 2, and recognized the importance of the discriminant.During the Renaissance in 1545, Gerolamo Cardano found the solution to equations of degree 3 and Lodovico Ferrarisolved equations of degree 4. Finally Niels Henrik Abel proved, in 1824, that equations of degree 5 and equations ofhigher degree are not always solvable using radicals. Galois theory, named after variste Galois, was introduced togive criteria deciding if an equation is solvable using radicals.

14

6.2. AREAS OF STUDY 15

6.2 Areas of studyThe algebraic equations are the basis of a number of areas of modern mathematics: Algebraic number theory is thestudy of (univariate) algebraic equations over the rationals. Galois theory has been introduced by variste Galois forgetting criteria deciding if an algebraic equation may be solved in terms of radicals. In eld theory, an algebraic exten-sion is an extension such that every element is a root of an algebraic equation over the base eld. Transcendence theoryis the study of the real numbers which are not solutions to an algebraic equation over the rationals. A Diophantineequation is a (usually multivariate) polynomial equation with integer coecients for which one is interested in theinteger solutions. Algebraic geometry is the study of the solutions in an algebraically closed eld of multivariatepolynomial equations.Two equations are equivalent if they have the same set of solutions. In particular the equation P = Q is equivalentwith P Q = 0 . It follows that the study of algebraic equations is equivalent to the study of polynomials.A polynomial equation over the rationals can always be converted to an equivalent one in which the coecients areintegers. For example, multiplying through by 42 = 237 and grouping its terms in the rst member, the previouslymentioned polynomial equation y4 + xy2 = x

3

3 xy2 + y2 17 becomes

42y4 + 21xy 14x3 + 42xy2 42y2 + 6 = 0:Because sine, exponentiation, and 1/T are not polynomial functions,

eTx2 +1

Txy + sin(T )z 2 = 0

is not a polynomial equation in the four variables x, y, z, and T over the rational numbers. However, it is a polynomialequation in the three variables x, y, and z over the eld of the elementary functions in the variable T.As for any equation, the solutions of an equation are the values of the variables for which the equation is true. Forunivariate algebraic equations these are also called roots, even if, properly speaking, one should say the solutions ofthe algebraic equation P=0 are the roots of the polynomial P. When solving an equation, it is important to specify inwhich set the solutions are allowed. For example, for an equation over the rationals onemay look for solutions in whichall the variables are integers. In this case the equation is a diophantine equation. One may also be interested only inthe real solutions. However, for univariate algebraic equations, the number of solutions is nite, and all solutions arecontained in any algebraically closed eld containing the coecientsfor example, the eld of complex numbers inthe case of equations over the rationals. It follows that without precision root and solution usually mean solutionin an algebraically closed eld.

6.3 See also Algebraic function Algebraic number Root nding Linear equation (degree = 1) Quadratic equation (degree = 2) Cubic equation (degree = 3) Quartic equation (degree = 4) Quintic equation (degree = 5) Sextic equation (degree = 6) Septic equation (degree = 7) System of linear equations

16 CHAPTER 6. ALGEBRAIC EQUATION

System of polynomial equations Linear Diophantine equation Linear equation over a ring Cramers theorem (algebraic curves), on the number of points usually sucient to determine a bivariate n-thdegree curve

6.4 References Hazewinkel, Michiel, ed. (2001), Algebraic equation, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Weisstein, Eric W., Algebraic Equation, MathWorld.

Chapter 7

Algebraic variety

This article is about algebraic varieties. For the term variety of algebras, and an explanation of the dierencebetween a variety of algebras and an algebraic variety, see variety (universal algebra).In mathematics, algebraic varieties (also called varieties) are one of the central objects of study in algebraic

The twisted cubic is a projective algebraic variety.

17

18 CHAPTER 7. ALGEBRAIC VARIETY

geometry. Classically, an algebraic variety was dened to be the set of solutions of a system of polynomial equations,over the real or complex numbers. Modern denitions of an algebraic variety generalize this notion in several dierentways, while attempting to preserve the geometric intuition behind the original denition.[1]:58

Conventions regarding the denition of an algebraic variety dier slightly. For example, some authors require thatan "algebraic variety" is, by denition, irreducible (which means that it is not the union of two smaller sets that areclosed in the Zariski topology), while others do not. When the former convention is used, non-irreducible algebraicvarieties are called algebraic sets.The notion of variety is similar to that of manifold, the dierence being that a variety may have singular points, whilea manifold will not. In many languages, both varieties and manifolds are named by the same word.Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra and geometry byshowing that a monic polynomial (an algebraic object) in one variable with complex coecients is determined by theset of its roots (a geometric object) in the complex plane. Generalizing this result, Hilberts Nullstellensatz providesa fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz andrelated results, mathematicians have established a strong correspondence between questions on algebraic sets andquestions of ring theory. This correspondence is the specicity of algebraic geometry among the other subareas ofgeometry.

7.1 Introduction and denitions

An ane variety over an algebraically closed eld is conceptually the easiest type of variety to dene, which will bedone in this section. Next, one can dene projective and quasi-projective varieties in a similar way. The most generaldenition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that onecan construct genuinely new examples of varieties in this way, but Nagata gave an example of such a new variety inthe 1950s.

7.1.1 Ane varieties

Main article: Ane variety

Let k be an algebraically closed eld and let An be an ane n-space over k. The polynomials f in the ring k[x1, ...,xn] can be viewed as k-valued functions on An by evaluating f at the points in An, i.e. by choosing values in A foreach xi. For each set S of polynomials in k[x1, ..., xn], dene the zero-locus Z(S) to be the set of points in An onwhich the functions in S simultaneously vanish, that is to say

Z(S) = fx 2 An j f(x) = 0 all for f 2 Sg :

A subset V of An is called an ane algebraic set if V = Z(S) for some S.[1]:2 A nonempty ane algebraic set Vis called irreducible if it cannot be written as the union of two proper algebraic subsets.[1]:3 An irreducible anealgebraic set is also called an ane variety.[1]:3 (Many authors use the phrase ane variety to refer to any anealgebraic set, irreducible or not[note 1])Ane varieties can be given a natural topology by declaring the closed sets to be precisely the ane algebraic sets.This topology is called the Zariski topology.[1]:2

Given a subset V of An, we dene I(V) to be the ideal of all polynomial functions vanishing on V :

I(V ) = ff 2 k[x1; ; xn] j f(x) = 0 all for x 2 V g :

For any ane algebraic set V, the coordinate ring or structure ring of V is the quotient of the polynomial ring bythis ideal.[1]:4

7.2. EXAMPLES 19

7.1.2 Projective varieties and quasi-projective varieties

Main articles: Projective variety and Quasi-projective variety

Let k be an algebraically closed eld and let Pn be the projective n-space over k. Let f in k[x0, ..., xn] be ahomogeneous polynomial of degree d. It is not well-dened to evaluate f on points in Pn in homogeneous coor-dinates. However, because f is homogeneous, f (x0, ..., xn) = d f (x0, ..., xn), it does make sense to ask whetherf vanishes at a point [x0 : ... : xn]. For each set S of homogeneous polynomials, dene the zero-locus of S to be theset of points in Pn on which the functions in S vanish:

Z(S) = fx 2 Pn j f(x) = 0 all for f 2 Sg:

A subset V of Pn is called a projective algebraic set if V = Z(S) for some S.[1]:9 An irreducible projective algebraicset is called a projective variety.[1]:10

Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed.Given a subset V of Pn, let I(V) be the ideal generated by all homogeneous polynomials vanishing on V. For anyprojective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.[1]:10

A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every ane variety is quasi-projective.[2] Notice also that the complement of an algebraic set in an ane variety is a quasi-projective variety; inthe context of ane varieties, such a quasi-projective variety is usually not called a variety but a constructible set.

7.1.3 Abstract varieties

In classical algebraic geometry, all varieties were by denition quasiprojective varieties, meaning that they were opensubvarieties of closed subvarieties of projective space. For example, in Chapter 1 of Hartshorne a variety over analgebraically closed eld is dened to be a quasi-projective variety,[1]:15 but from Chapter 2 onwards, the term variety(also called an abstract variety) refers to a more general object, which locally is a quasi-projective variety, but whenviewed as a whole is not necessarily quasi-projective; i.e. it might not have an embedding into projective space.[1]:105So classically the denition of an algebraic variety required an embedding into projective space, and this embeddingwas used to dene the topology on the variety and the regular functions on the variety. The disadvantage of sucha denition is that not all varieties come with natural embeddings into projective space. For example, under thisdenition, the product P1 P1 is not a variety until it is embedded into the projective space; this is usually done bythe Segre embedding. However, any variety that admits one embedding into projective space admits many others bycomposing the embedding with the Veronese embedding. Consequently many notions that should be intrinsic, suchas the concept of a regular function, are not obviously so.The earliest successful attempt to dene an algebraic variety abstractly, without an embedding, was made by AndrWeil. In his Foundations of Algebraic Geometry, Weil dened an abstract algebraic variety using valuations. ClaudeChevalley made a denition of a scheme, which served a similar purpose, but was more general. However, it wasAlexander Grothendieck's denition of a scheme that was both most general and found the most widespread accep-tance. In Grothendiecks language, an abstract algebraic variety is usually dened to be an integral, separated schemeof nite type over an algebraically closed eld,[note 2] although some authors drop the irreducibility or the reducednessor the separateness condition or allow the underlying eld to be not algebraically closed.[note 3] Classical algebraicvarieties are the quasiprojective integral separated nite type schemes over an algebraically closed eld.

Existence of non-quasiprojective abstract algebraic varieties

One of the earliest examples of a non-quasiprojective algebraic variety were given by Nagata.[3] Nagatas examplewas not complete (the analog of compactness), but soon afterwards he found an algebraic surface that was completeand non-projective.[4] Since then other examples have been found.

7.2 Examples


7.2.1 SubvarietyA subvariety is a subset of a variety that is itself a variety (with respect to the structure induced from the ambientvariety). For example, every open subset of a variety is a variety. See also closed immersion.Hilberts Nullstellensatz says that closed subvarieties of an ane or projective variety are in one-to-one correspondencewith the prime ideals or homogeneous prime ideals of the coordinate ring of the variety.

7.2.2 Ane varietyExample 1

Let k = C, and A2 be the two-dimensional ane space over C. Polynomials in the ring C[x, y] can be viewed ascomplex valued functions on A2 by evaluating at the points in A2. Let subset S of C[x, y] contain a single element f(x, y):

f(x; y) = x+ y 1:

The zero-locus of f (x, y) is the set of points inA2 on which this function vanishes: it is the set of all pairs of complexnumbers (x, y) such that y = 1 x, commonly known as a line. This is the set Z( f ):

Z(f) = f(x; 1 x) 2 C2g:

Thus the subset V = Z( f ) of A2 is an algebraic set. The set V is not empty. It is irreducible, as it cannot be writtenas the union of two proper algebraic subsets. Thus it is an ane algebraic variety.

Example 2

Let k = C, and A2 be the two-dimensional ane space over C. Polynomials in the ring C[x, y] can be viewed ascomplex valued functions on A2 by evaluating at the points in A2. Let subset S of C[x, y] contain a single elementg(x, y):

g(x; y) = x2 + y2 1:

The zero-locus of g(x, y) is the set of points in A2 on which this function vanishes, that is the set of points (x,y) suchthat x2 + y2 = 1. As g(x, y) is an absolutely irreducible polynomial, this is an algebraic variety. The set of its realpoints (that is the points for which x and y are real numbers), is known as the unit circle; this name is also often givento the whole variety.

Example 3

The following example is neither a hypersurface, nor a linear space, nor a single point. LetA3 be the three-dimensionalane space over C. The set of points (x, x2, x3) for x in C is an algebraic variety, and more precisely an algebraiccurve that is not contained in any plane.[note 4] It is the twisted cubic shown in the above gure. It may be dened bythe equations

y x2 = 0z x3 = 0

The fact that the set of the solutions of this system of equations is irreducible needs a proof. The simplest results fromthe fact that the projection (x, y, z) (x, y) is injective on the set of the solutions and that its image is an irreducibleplane curve.

7.2. EXAMPLES 21

For more dicult examples, a similar proof may always be given, but may imply a dicult computation: rst aGrbner basis computation to compute the dimension, followed by a random linear change of variables (not alwaysneeded); then a Grbner basis computation for another monomial ordering to compute the projection and to provethat it is injective, and nally a polynomial factorization to prove the irreducibility of the image.

7.2.3 Projective varietyA projective variety is a closed subvariety of a projective space. That is, it is the zero locus of a set of homogeneouspolynomials that generate a prime ideal.

Example 1

The ane plane curve y2 = x3 - x. The corresponding projective curve is called an elliptic curve.

A plane projective curve is the zero locus of an irreducible homogeneous polynomial in three indeterminates. Theprojective lineP1 is an example of a projective curve, since it appears as the zero locus of one homogeneous coordinatein the projective plane. For another example, rst consider the ane cubic curve:

y2 = x3 x


in the 2-dimensional ane space (over a eld of characteristic not two). It has the associated cubic homogeneouspolynomial equation:

y2z = x3 xz2

which denes a curve in P2 called an elliptic curve. The curve has genus one (genus formula); in particular, it is notisomorphic to the projective line P1, which has genus zero. Using genus to distinguish curves is very basic: in fact,the genus is the rst invariant one uses to classify curves (see also the construction of moduli of algebraic curves).

Example 2

LetV be a nite-dimensional vector space. The Grassmannian varietyGn(V) is the set of all n-dimensional subspacesof V. It is a projective variety: it is embedded into a projective space via the Plcker embedding:

Gn(V ) ,! P(^nV ); hb1; : : : ; bni 7! [b1 ^ ^ bn]

where bi are any set of linearly independent vectors in V, ^nV is the n-th exterior power of V and the bracket [w]means the line spanned by the nonzero vector w.TheGrassmannian variety comeswith a natural vector bundle (or locally free sheaf to be precise) called the tautologicalbundle, which is important in the study of characteristic classes such as Chern classes.

7.3 Basic results An ane algebraic set V is a variety if and only if I(V) is a prime ideal; equivalently, V is a variety if and onlyif its coordinate ring is an integral domain.[5]:52[1]:4

Every nonempty ane algebraic set may be written uniquely as a nite union of algebraic varieties (where noneof the varieties in the decomposition is a subvariety of any other).[1]:5

The dimension of a variety may be dened in various equivalent ways. See Dimension of an algebraic varietyfor details.

7.4 Isomorphism of algebraic varietiesSee also: morphism of varieties

Let V1, V2 be algebraic varieties. We say V1 and V2 are isomorphic, and write V1 V2, if there are regular maps : V1 V2 and : V2 V1 such that the compositions and are the identity maps on V1 and V2respectively.

7.5 Discussion and generalizationsThe basic denitions and facts above enable one to do classical algebraic geometry. To be able to do more forexample, to deal with varieties over elds that are not algebraically closed some foundational changes are required.The modern notion of a variety is considerably more abstract than the one above, though equivalent in the case ofvarieties over algebraically closed elds. An abstract algebraic variety is a particular kind of scheme; the generalizationto schemes on the geometric side enables an extension of the correspondence described above to a wider class ofrings. A scheme is a locally ringed space such that every point has a neighbourhood that, as a locally ringed space,is isomorphic to a spectrum of a ring. Basically, a variety over k is a scheme whose structure sheaf is a sheaf ofk-algebras with the property that the rings R that occur above are all integral domains and are all nitely generatedk-algebras, that is to say, they are quotients of polynomial algebras by prime ideals.

7.6. ALGEBRAIC MANIFOLDS 23

This denition works over any eld k. It allows you to glue ane varieties (along common open sets) without worryingwhether the resulting object can be put into some projective space. This also leads to diculties since one canintroduce somewhat pathological objects, e.g. an ane line with zero doubled. Such objects are usually not consideredvarieties, and are eliminated by requiring the schemes underlying a variety to be separated. (Strictly speaking, thereis also a third condition, namely, that one needs only nitely many ane patches in the denition above.)Some modern researchers also remove the restriction on a variety having integral domain ane charts, and whenspeaking of a variety only require that the ane charts have trivial nilradical.A complete variety is a variety such that any map from an open subset of a nonsingular curve into it can be extendeduniquely to the whole curve. Every projective variety is complete, but not vice versa.These varieties have been called 'varieties in the sense of Serre', since Serre's foundational paper FAC on sheafcohomology was written for them. They remain typical objects to start studying in algebraic geometry, even if moregeneral objects are also used in an auxiliary way.One way that leads to generalisations is to allow reducible algebraic sets (and elds k that aren't algebraically closed),so the rings R may not be integral domains. A more signicant modication is to allow nilpotents in the sheaf ofrings. A nilpotent in a eld must be 0: these if allowed in coordinate rings aren't seen as coordinate functions.From the categorical point of view, nilpotents must be allowed, in order to have nite limits of varieties (to get berproducts). Geometrically this says that bres of good mappings may have 'innitesimal' structure. In the theory ofschemes of Grothendieck these points are all reconciled: but the general scheme is far from having the immediategeometric content of a variety.There are further generalizations called algebraic spaces and stacks.

7.6 Algebraic manifoldsMain article: Algebraic manifold

An algebraic manifold is an algebraic variety that is also an m-dimensional manifold, and hence every sucientlysmall local patch is isomorphic to km. Equivalently, the variety is smooth (free from singular points). When k is thereal numbers, R, algebraic manifolds are called Nash manifolds. Algebraic manifolds can be dened as the zero setof a nite collection of analytic algebraic functions. Projective algebraic manifolds are an equivalent denition forprojective varieties. The Riemann sphere is one example.

7.7 See also Variety (disambiguation) listing also several mathematical meanings

Function eld of an algebraic variety

Dimension of an algebraic variety

Singular point of an algebraic variety

Birational geometry

Abelian variety

Motive

Scheme

Analytic variety

ZariskiRiemann space

Semi-algebraic set


7.8 Footnotes[1] Hartshorne, p.xv, notes that his choice is not conventional; see for example, Harris, p.3

[2] Hartshorne 1976, pp. 104105

[3] Liu, Qing. Algebraic Geometry and Arithmetic Curves, p. 55 Denition 2.3.47, and p. 88 Example 3.2.3

[4] Harris, p.9; that it is irreducible is stated as an exercise in Hartshorne p.7

7.9 References[1] Hartshorne, Robin (1977). Algebraic Geometry. Springer-Verlag. ISBN 0-387-90244-9.

[2] Hartshorne, Exercise I.2.9, p.12

[3] Nagata, Masayoshi (1956), On the imbedding problem of abstract varieties in projective varieties,Memoirs of the Collegeof Science, University of Kyoto. Series A: Mathematics 30: 7182, MR 0088035

[4] Nagata, Masayoshi (1957), On the imbeddings of abstract surfaces in projective varieties, Memoirs of the College ofScience, University of Kyoto. Series A: Mathematics 30: 231235, MR 0094358

[5] Harris, Joe (1992). Algebraic Geometry - A rst course. Springer-Verlag. ISBN 0-387-97716-3.

Cox, David; John Little; Don O'Shea (1997). Ideals, Varieties, and Algorithms (second ed.). Springer-Verlag.ISBN 0-387-94680-2.

Eisenbud, David (1999). Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag.ISBN 0-387-94269-6.

Milne, James S. (2008). Algebraic Geometry. Retrieved 2009-09-01.

This article incorporates material from Isomorphism of varieties on PlanetMath, which is licensed under the CreativeCommons Attribution/Share-Alike License.

Chapter 8

All one polynomial

An all one polynomial (AOP) is a polynomial in which all coecients are one. Over the nite eld of order two, con-ditions for the AOP to be irreducible are known, which allow this polynomial to be used to dene ecient algorithmsand circuits for multiplication in nite elds of characteristic two.[1] The AOP is a 1-equally spaced polynomial.[2]

8.1 DenitionAn AOP of degree m has all terms from xm to x0 with coecients of 1, and can be written as

AOPm(x) =mXi=0

xi

or

AOPm(x) = xm + xm1 + + x+ 1

or

AOPm(x) =xm+1 1x 1

thus the roots of the all one polynomial of degree m are all (m+1)th roots of unity other than unity itself.

8.2 PropertiesOver GF(2) the AOP has many interesting properties, including:

The Hamming weight of the AOP is m + 1, the maximum possible for its degree[3]

The AOP is irreducible if and only if m + 1 is prime and 2 is a primitive root modulo m + 1[1]

The only AOP that is a primitive polynomial is x2 + x + 1.

Despite the fact that the Hamming weight is large, because of the ease of representation and other improvement

Documents

Polynomials ABC