Square-Root Parametrizations

IntroductionSquare-root Parametrization

Further Directions

Square-Root Parametrizations

Jose Manuel Garcia Vallinas / Josef Schicho

Spezialforschungsbereich F013Subproject F1303

Johann Radon Institute for Computional and Applied Mathematics (RICAM)Austrian Academy of Sciences (OAW)

Linz, Austria

Workshop on Algebraic Spline Curves and SurfacesMay 17-18, 2006, Eger, Hungary

Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations


Further Directions

Definitions IDefinitions IIWeierstrass Form

Definitions I

Plane Algebraic Curve

An affine irreducible plane algebraic curve over C is defined asthe set

C = {(a, b) ∈ A2(C)|f (a, b) = 0}

for a non-constant irreducible polynomial f (x , y) ∈ C[x , y ].

Singular Point

Let C be an affine plane curve over C defined by f (x , y) ∈ C[x , y ]and let P = (a, b) ∈ C . P is a singular point if and only if theorder of the first non-vanishing term in the Taylor expansion of fat P is greater than 1.



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Definitions I

Plane Algebraic Curve

An affine irreducible plane algebraic curve over C is defined asthe set

C = {(a, b) ∈ A2(C)|f (a, b) = 0}

for a non-constant irreducible polynomial f (x , y) ∈ C[x , y ].

Singular Point

Let C be an affine plane curve over C defined by f (x , y) ∈ C[x , y ]and let P = (a, b) ∈ C . P is a singular point if and only if theorder of the first non-vanishing term in the Taylor expansion of fat P is greater than 1.



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Definitions II

GenusLet C be an irreducible plane curve of degree n, having onlyordinary singularities of multiplicities r1, . . . , rm. The genus of C ,g(C ) , is defined as

g(C ) :=1

2[(n − 1)(n − 2)−

m∑i=1

ri (ri − 1)]

For non-ordinary singularities, the genus can be computed similarly.



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Weierstrass Form

A curve C is called elliptic if and only if its genus is 1

It is known that an elliptic curve can be birationally transformed tothe form y2 = F (x), where F (x) is a square-free polynomial in xof degree 3.

A curve C is called hyper-elliptic if and only if its genus g isgreater than 1 and it can be birationally transformed to y2 = F (x),where F (x) is a polynomial in x of degree 2g + 1 or 2g + 2.



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Weierstrass Form






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Weierstrass Form






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Definition and ProblemClassification

Square-root Parametrization Problem

Square-root Parameterisable Curve

A curve is called square-root parameterisable if it can beparameterised in terms of t and

√P(t), where P(t) is a

polynomial in t.

Example: x2 + y2 − 1: t 7→ (t,√

1− t2) or (√

1− t2, t)

ProblemWe have an algebraic (irreducible) plane curve C and we want toknow if it is square-root parameterisable and compute thisparametrization in the positive case.



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polynomial in t.

Example: x2 + y2 − 1:

t 7→ (t,√

1− t2) or (√

1− t2, t)




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polynomial in t.

Example: x2 + y2 − 1: t 7→ (t,√

1− t2) or (√

1− t2, t)




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polynomial in t.

Example: x2 + y2 − 1: t 7→ (t,√

1− t2) or (√

1− t2, t)




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Rational Curves

If we have a rational curve, we take its rational parametrization.

Example

frational := y2 − x2(x + 1);genus(frational , x , y) = 0;squareRoot(frational , x , y , t) = [−1 + t2, t(−1 + t2)]



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Rational Curves


Example

frational := y2 − x2(x + 1);

genus(frational , x , y) = 0;squareRoot(frational , x , y , t) = [−1 + t2, t(−1 + t2)]



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Rational Curves


Example

frational := y2 − x2(x + 1);genus(frational , x , y) = 0;

squareRoot(frational , x , y , t) = [−1 + t2, t(−1 + t2)]



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Rational Curves


Example

frational := y2 − x2(x + 1);genus(frational , x , y) = 0;squareRoot(frational , x , y , t) = [−1 + t2, t(−1 + t2)]



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Elliptic Curves

If we have an elliptic curve, we simply need to compute itsWeierstrass form and substitute.

Example

felliptic := x3 − y3 − x ;genus(felliptic , x , y) = 1;WeierstrassForm(felliptic) = x3 + y2 − 1

squareRoot(felliptic , x , y , t) = [√−t3+1(t+1)3+3t+3t2 ,

√−t3+1(t+2)3+3t+3t2 ]



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Elliptic Curves


Example

felliptic := x3 − y3 − x ;

genus(felliptic , x , y) = 1;WeierstrassForm(felliptic) = x3 + y2 − 1


√−t3+1(t+2)3+3t+3t2 ]



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Elliptic Curves


Example

felliptic := x3 − y3 − x ;genus(felliptic , x , y) = 1;

WeierstrassForm(felliptic) = x3 + y2 − 1


√−t3+1(t+2)3+3t+3t2 ]



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Elliptic Curves


Example



√−t3+1(t+2)3+3t+3t2 ]



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Elliptic Curves


Example



√−t3+1(t+2)3+3t+3t2 ]



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Hyper-elliptic Curves

In the case of a hyper-elliptic curve, we already have theparametrization. Simply compute the Weierstrass form of thecurve.

Example

fhyperelliptic := x4 + y4 + x2 + y2;genus(fhyperelliptic , x , y) = 2;WeierstrassForm(fhyperelliptic) = y2 + x6 + x2 + x4 + 1

squareRoot(fhyperelliptic , x , y , t) = [ t2+1√−t6−t4−t2−1

, (t2+1)t√−t6−t4−t2−1

]



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Example

fhyperelliptic := x4 + y4 + x2 + y2;

genus(fhyperelliptic , x , y) = 2;WeierstrassForm(fhyperelliptic) = y2 + x6 + x2 + x4 + 1


, (t2+1)t√−t6−t4−t2−1

]



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Example

fhyperelliptic := x4 + y4 + x2 + y2;genus(fhyperelliptic , x , y) = 2;

WeierstrassForm(fhyperelliptic) = y2 + x6 + x2 + x4 + 1


, (t2+1)t√−t6−t4−t2−1

]



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Example



, (t2+1)t√−t6−t4−t2−1

]



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Example



, (t2+1)t√−t6−t4−t2−1

]



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Non-hyperelliptic Curves

A non-hyperelliptic curve is a curve of genus greater than 2 whichis not hyperelliptic.

If the curve is non-hyperelliptic, it can be shown that there doesnot exist a square-root parametrization.

Example

fnon−hyperelliptic := x3 − y5 + y4 − y2;genus(fnon−hyperelliptic , x , y) = 3;squareRoot(fnon−hyperelliptic , x , y , t);Error, It is not square-root parameterisable.



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Example




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Example

fnon−hyperelliptic := x3 − y5 + y4 − y2;

genus(fnon−hyperelliptic , x , y) = 3;squareRoot(fnon−hyperelliptic , x , y , t);Error, It is not square-root parameterisable.



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Example

fnon−hyperelliptic := x3 − y5 + y4 − y2;genus(fnon−hyperelliptic , x , y) = 3;

squareRoot(fnon−hyperelliptic , x , y , t);Error, It is not square-root parameterisable.



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Example




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Further DirectionsReferences

Further Directions

I Study n-root parametrizations.

I Devise and implement algorithms for computing n-rootparametrizations.



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References

D. Eisenbud, Commutative Algebra with a view toward AlgebraicGeometry, Graduate Texts in Mathematics 150, Springer BerlinHeidelberg New York, 1995.

R. Hartshorne, Algebraic Geometry, Graduate Texts inMathematics,Springer Berlin Heidelberg New York, 1977

I.R. Shafarevich, Basic Algebraic Geometry ,Springer BerlinHeidelberg New York, 1974


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Square-Root Parametrizations