A homogeneous generalization of a theorem of Ganong and Daiglepjedrzej/research/Halifax2013.pdf · A homogeneous generalization of a theorem of Ganong and Daigle PiotrJędrzejewicz

A homogeneous generalizationof a theorem of Ganong and Daigle

Piotr Jędrzejewicz

Nicolaus Copernicus University

Toruń, Poland

2013 CMS Summer Meeting, Halifax, Nowa Szkocja

Piotr Jędrzejewicz On a theorem of Ganong and Daigle

Motivation

Theorem [Ganong, Daigle].

Let k be a field of characteristic p > 0, let A and R be polynomialk-algebras in two variables such that Ap $ R $ A, whereAp = {ap, a ∈ A}. Then there exist x , y ∈ A such that A = k[x , y ]and R = k[x , yp].

The above theorem was proved by Ganong in the case ofalgebraically closed field, and by Daigle in the general case.

R. Ganong, Plane Frobenius sandwiches, Proc. Amer. Math. Soc. 84(1982), 474–478.D. Daigle, Plane Frobenius sandwiches of degree p, Proc. Amer. Math.Soc. 117 (1993), 885–889.


Theorem A

In a joint work with Nowicki we generalized the above theorem to nvariables in the restriction to homogeneous case.

Let p be a prime number. Let k be a field (of arbitrarycharacteristic) and let f1, . . . , fn ∈ k[x1, . . . , xn] be homogeneouspolynomials such that k[xp

1 , . . . , xpn ] ⊆ k[f1, . . . , fn].

a) If char k 6= p, then k[f1, . . . , fn] = k[x i11 , . . . , x

inn ] for some

i1, . . . , in ∈ {1, p}.b) If char k = p, then k[f1, . . . , fn] = k[y1, . . . , ym, y

pm+1, . . . , y

pn ]

for some m ∈ {0, 1, . . . , n} and some k-linear basis y1, . . . , yn of〈x1, . . . , xn〉k .

A. Nowicki, PJ, Polynomial graded subalgebras of polynomial algebras,Comm. Algebra 40 (2012), 2853–2866.


ExampleIf k is a field of characteristic p > 0, then

k[xp, yp] ⊆ k[x + y , yp] ⊆ k[x , y ].

We have:xp = (x + y)p − yp,

soxp ∈ k[x + y , yp].


ExampleIf k is a field of characteristic p > 0, then

k[xp, yp] ⊆ k[x + y , yp] ⊆ k[x , y ].

We have:xp = (x + y)p − yp,

soxp ∈ k[x + y , yp].


The main tool in the proof of Theorem A:

Theorem [Krull].

Let R be a Noetherian commutative ring with unity, let P be aminimal prime ideal of an ideal generated by n elements. Then theheight of P is not greater than n.

However, some special cases of Theorem A:– n = 1, 2 (arbitrary p),– p = 2, 3 (arbitrary n),– n = 3, p 6 19,– n = 4, p 6 7,were obtained earlier using the following fact.


The main tool in the proof of Theorem A:

Theorem [Krull].

Let R be a Noetherian commutative ring with unity, let P be aminimal prime ideal of an ideal generated by n elements. Then theheight of P is not greater than n.

However, some special cases of Theorem A:– n = 1, 2 (arbitrary p),– p = 2, 3 (arbitrary n),– n = 3, p 6 19,– n = 4, p 6 7,were obtained earlier using the following fact.


Proposition.Denote by r1, . . . , rn the degrees of f1, . . . , fn, respectively. Assumethat the statement holds for n − 1 and the equation

r1x1 + . . .+ rn−1xn−1 = p

has less than n − 1 solutions in non-negative integers. Then thestatement holds for n.

PJ, On polynomial graded subalgebras of a polynomial algebra, AlgebraColloquium 17 (2010), 929–936.


Some combinatorical aspects

Let k be a field. Let g1, . . . , gn ∈ k[x1, . . . , xn] be algebraicallyindependent homogeneous polynomials of degrees s1, . . . , sn,respectively.

We are looking for subalgebras R = k[f1, . . . , fn] such that

k[g1, . . . , gn] ⊆ R,

where f1, . . . , fn are homogeneous polynomials of degrees r1, . . . , rn,respectively.

Problem I.Describe all possible degrees r1, . . . , rn.

Problem II.Given r1, . . . , rn, describe all such subalgebras R .





k[g1, . . . , gn] ⊆ R,








k[g1, . . . , gn] ⊆ R,








k[g1, . . . , gn] ⊆ R,





An application to derivations

A k-linear map d : k[x1, . . . , xn]→ k[x1, . . . , xn] such that

d(fg) = fd(g) + gd(f )

for all f , g ∈ k[x1, . . . , xn] is called a k-derivation.

The unique k-derivation of k[x1, . . . , xn] such that

d(x1) = g1, . . . , d(xn) = gn

is of the form d = g1 · ∂∂x1

+ . . .+ gn · ∂∂xn

.

The kernel of a k-derivation d

k[x1, . . . , xn]d = {f ∈ k[x1, . . . , xn] : d(f ) = 0}

is a k-subalgebra, called the ring of constants of d .

If char k = p > 0, then k[xp1 , . . . , x

pn ] ⊆ k[x1, . . . , xn]

d .







d(x1) = g1, . . . , d(xn) = gn


+ . . .+ gn · ∂∂xn

.


k[x1, . . . , xn]d = {f ∈ k[x1, . . . , xn] : d(f ) = 0}



pn ] ⊆ k[x1, . . . , xn]

d .







d(x1) = g1, . . . , d(xn) = gn


+ . . .+ gn · ∂∂xn

.


k[x1, . . . , xn]d = {f ∈ k[x1, . . . , xn] : d(f ) = 0}



pn ] ⊆ k[x1, . . . , xn]

d .







d(x1) = g1, . . . , d(xn) = gn


+ . . .+ gn · ∂∂xn

.


k[x1, . . . , xn]d = {f ∈ k[x1, . . . , xn] : d(f ) = 0}



pn ] ⊆ k[x1, . . . , xn]

d .


A k-derivation d of k[x1, . . . , xn] is called homogeneous of degree rif d(x1), . . . , d(xn) are homogeneous of degree r + 1. In this case,for every homogeneous polynomial f of degree s, the polynomiald(f ) is homogeneous of degree r + s. The ring of constants of ahomogeneous derivation is a graded subalgebra.

Corollary 1.

Let d be a homogeneous k-derivation of k[x1, . . . , xn], where k is afield of characteristic p > 0. Then k[x1, . . . , xn]

d is a polynomialk-algebra if and only if

(∗) k[x1, . . . , xn]d = k[y1, . . . , ym, y

pm+1, . . . , y

pn ]



A k-derivation d of k[x1, . . . , xn] is called homogeneous of degree rif d(x1), . . . , d(xn) are homogeneous of degree r + 1. In this case,for every homogeneous polynomial f of degree s, the polynomiald(f ) is homogeneous of degree r + s. The ring of constants of ahomogeneous derivation is a graded subalgebra.

Corollary 1.

Let d be a homogeneous k-derivation of k[x1, . . . , xn], where k is afield of characteristic p > 0. Then k[x1, . . . , xn]

d is a polynomialk-algebra if and only if

(∗) k[x1, . . . , xn]d = k[y1, . . . , ym, y

pm+1, . . . , y

pn ]



A homogeneous k-derivation d of k[x1, . . . , xn] of degree 0 is calledlinear. In this case a restriction of d to 〈x1, . . . , xn〉k is a k-linearendomorphism.

Corollary 2.

Let d be a linear derivation of k[x1, . . . , xn], where k is a field ofcharacteristic p > 0. Then k[x1, . . . , xn]

d is a polynomial k-algebraif and only if the Jordan form of d |〈x1,...,xn〉k is one of the following:

ρ1 0. . .

0 ρn

,

(ρ1 10 ρ1

)0

ρ2. . .

0 ρn−1

,

ρ1 1 00 ρ1 10 0 ρ1

0

ρ2. . .

0 ρn−2

︸︷︷︸

only p = 2

,

where nonzero ρi are linearly independent over Fp.


A homogeneous k-derivation d of k[x1, . . . , xn] of degree 0 is calledlinear. In this case a restriction of d to 〈x1, . . . , xn〉k is a k-linearendomorphism.

Corollary 2.

Let d be a linear derivation of k[x1, . . . , xn], where k is a field ofcharacteristic p > 0. Then k[x1, . . . , xn]

d is a polynomial k-algebraif and only if the Jordan form of d |〈x1,...,xn〉k is one of the following:

ρ1 0. . .

0 ρn

,

(ρ1 10 ρ1

)0

ρ2. . .

0 ρn−1

,

ρ1 1 00 ρ1 10 0 ρ1

0

ρ2. . .

0 ρn−2

︸︷︷︸

only p = 2

,

where nonzero ρi are linearly independent over Fp.


Theorem B

Let k be a field and let f1, . . . , fn ∈ k[x1, . . . , xn] be homogeneouspolynomials such that

k[xp11 , . . . , xpn

n ] ⊆ k[f1, . . . , fn]

for some prime numbers p1, . . . , pn.

a) If char k 6∈ {p1, . . . , pn}, then k[f1, . . . , fn] = k[x i11 , . . . , x

inn ] for

some i1 ∈ {1, p1}, . . . , in ∈ {1, pn}.b) If char k ∈ {p1, . . . , pn}, then k[f1, . . . , fn] = k[y i1

1 , . . . , yinn ] for

some i1 ∈ {1, p1}, . . . , in ∈ {1, pn} and some k-linear basisy1, . . . , yn of 〈x1, . . . , xn〉k such that yi = xi if pi 6= char k ,

〈yi ; i ∈ T 〉k = 〈xi ; i ∈ T 〉k ,

where T = {i ∈ {1, . . . , n}; char k = pi}.


Theorem B


k[xp11 , . . . , xpn

n ] ⊆ k[f1, . . . , fn]

for some prime numbers p1, . . . , pn.a) If char k 6∈ {p1, . . . , pn}, then k[f1, . . . , fn] = k[x i1

1 , . . . , xinn ] for

some i1 ∈ {1, p1}, . . . , in ∈ {1, pn}.

b) If char k ∈ {p1, . . . , pn}, then k[f1, . . . , fn] = k[y i11 , . . . , y

inn ] for


〈yi ; i ∈ T 〉k = 〈xi ; i ∈ T 〉k ,

where T = {i ∈ {1, . . . , n}; char k = pi}.


Theorem B


k[xp11 , . . . , xpn

n ] ⊆ k[f1, . . . , fn]

for some prime numbers p1, . . . , pn.a) If char k 6∈ {p1, . . . , pn}, then k[f1, . . . , fn] = k[x i1

1 , . . . , xinn ] for

some i1 ∈ {1, p1}, . . . , in ∈ {1, pn}.b) If char k ∈ {p1, . . . , pn}, then k[f1, . . . , fn] = k[y i1

1 , . . . , yinn ] for


〈yi ; i ∈ T 〉k = 〈xi ; i ∈ T 〉k ,

where T = {i ∈ {1, . . . , n}; char k = pi}.Piotr Jędrzejewicz On a theorem of Ganong and Daigle

Two special cases.

Let k be a field, let f1, . . . , fn ∈ k[x1, . . . , xn] be homogeneouspolynomials such that

k[xp11 , . . . , xpn

n ] ⊆ k[f1, . . . , fn]


Then k[f1, . . . , fn] = k[x i11 , . . . , x

inn ] for some

i1 ∈ {1, p1}, . . . , in ∈ {1, pn}, if:

– k is a field of characteristic 0,

or

– p1, . . . , pn are pairwise different.


Two special cases.


k[xp11 , . . . , xpn

n ] ⊆ k[f1, . . . , fn]


Then k[f1, . . . , fn] = k[x i11 , . . . , x

inn ] for some

i1 ∈ {1, p1}, . . . , in ∈ {1, pn}, if:


or



Two special cases.


k[xp11 , . . . , xpn

n ] ⊆ k[f1, . . . , fn]


Then k[f1, . . . , fn] = k[x i11 , . . . , x

inn ] for some

i1 ∈ {1, p1}, . . . , in ∈ {1, pn}, if:


or



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A homogeneous generalization of a theorem of Ganong and Daiglepjedrzej/research/Halifax2013.pdf · A homogeneous generalization of a theorem of Ganong and Daigle PiotrJędrzejewicz