Newton polytopes for constants of locally nilpotent derivations in polynomial rings

Newton polytopes for constants of locally nilpotent derivations in polynomial rings

Andrzej Nowicki

Nicolaus Copernicus University,

Faculty of Mathematics and Computer Science, 87-100 Toru´ n, Poland, (e-mail: anow@mat.uni.torun.pl).

Seminar Derivations and Automorphisms, 16 October, 1998

Abstract

Let d be a nonzero locally nilpotent derivation of a polynomial ring and let g be a constant of d. We prove that every vertex of the Newton polytopes of g has at least one zero coordinate.

1 Introduction

Let k be a field of characteristic zero and k[X] = k[x₁, . . . , x_n] the polynomial ring in n variables over k. If d is a derivation of k[X] then we denote by k[X]^d the ring of constants of d, that is, k[X]^d= Kerd = {w ∈ k[X]; d(w) = 0}.

In this note we prove the following

Theorem 1.1. Let d : k[X] → k[X] be a nonzero locally nilpotent derivation and let g ∈ k[X]^d. Then every vertex of the Newton polytopes of g has at least one zero coordinate.

In particular, if g is a constant of a nonzero locally nilpotent derivation of k[x, y], then (by the above theorem) the Newton polygon of g is either a point or a segment or a right triangle in R².

The proof is mainly based on the papers of Hadas [4], [3] and Makar - Limanov [8], [9].

2 Definitions, notations and basic facts

Throughout the paper n > 2 is a fixed natural number and Ω = Ωn denotes the set of all sequences of the form (α₁, . . . , α_n), where α₁, . . . , α_n are non-negative integers.

If α = (α₁, . . . , α_n) ∈ Ω, then X^α denotes the monomial x^α₁¹· · · x^α_nⁿ.

Let 0 6= f ∈ k[X]. We denote by Sf the support of f , that is, Sf is the set of all points α ∈ Ω such that the monomial X^α appears in f with a nonzero coefficient. We denote by N_f the convex hull (in the real space Rⁿ) of S_f ∪ {0}. The set N_f is called

the Newton polytope of f . If n = 2, then N_f is a Newton polygon. Many properties and applications of Newton polygons one can find, for example, in [1], [7], [10], [11].

Let p = (p₁, . . . , p_n) be a fixed nonzero point belonging to Rⁿ. If α = (α₁, . . . , α_n) ∈ Ω then pα denotes the sum p₁α₁+ · · · + p_nα_n. A nonzero polynomial f ∈ k[X] is said to be a p-form of degree s if f is of the form f = P

pα=sa_αX^α, where a_α ∈ k. We assume that the zero polynomial is a p-form of any degree.

Denote by A^(s)p the set of all p-forms of degree s. Each A^(s)p is a k-subspace of k[X]

and k[X] = L

r∈RA^(s)p . Moreover, A^(s)p A^(t)p ⊆ A^(s+t)p for all s, t ∈ R. Thus, k[X] is a graded ring. Such a gradation on k[X] is said to be a p-gradation.

Every polynomial f ∈ k[X] has the p-decomposition f =P f^(s) into p-components f^(s) of degree s. If f 6= 0, then deg_p(f ) denotes the p-degree of f , that is, the maximal s ∈ R such that f^(s) 6= 0. We denote by f^∗ the p-component of f of the maximal degree. We assume also that 0^∗ = 0.

If the numbers p₁, . . . , p_n are linearly independent over Q, then we say that the point p = (p₁, . . . , p_n) is primary. In this case p-forms are monomials.

Proposition 2.1 ([4]). Assume that p = (p1, . . . , pn) is a primary point of Rⁿ and f₁, . . . , f_s (where s > 1) are algebraically independent over k polynomials from k[X].

Let A = k[f₁, . . . , f_s]. Then there exist polynomials Q₁, . . . , Q_s belonging to A such that (a) Q₁ = f₁ and

(b) the p-forms Q^∗₁, . . . , Q^∗_s are algebraically independent over k.

The proof is the exact repetition of the arguments given in [4] (Proposition 3.2).  Let us recall that a k-linear mapping d : k[X] → k[X] is said to be a derivation of k[X] if d(f g) = f d(g) + d(f )g for all f, g ∈ k[X]. Note that a derivation d of k[X]

is completely defined by its values on the variables x₁, . . . , x_n. If (f₁, . . . , f_n) ∈ k[X]ⁿ, then there exists a unique derivation d of k[X] such that d(x_i) = f_i, for i = 1, . . . , n.

This derivation is defined by d(ϕ) =Pn

i=1f_i(∂ϕ/∂x_i), for all ϕ ∈ k[X].

Let d : k[X] → k[X] be a derivation. We say that d is locally nilpotent if for any f ∈ k[X] there exists m ∈ N such that d^m(f ) = 0. We say that d is p-homogeneous of degree r if

d A^(s)_p
⊆ A^(s+r)_p , for all s ∈ R.

It is easy to check that d is p-homogeneous of degree r if and only if d(x_i) ∈ A^(pp^i+r) for all i = 1, . . . , n.

Assume that d : k[X] → k[X] is a derivation and f₁, . . . , f_n are polynomials from k[X] such that d(x₁) = f₁, . . . , d(x_n) = f_n. Let 0 6= p ∈ Rⁿ and let r ∈ R. We denote by d_[r] the derivation of k[X] defined as:

d_[r](x_i) = f_i^(pi+r), i = 1, . . . , n.

This derivation is p-homogeneous of degree r. It is clear that d =P

r∈Rd_[r],

where d_[r] = 0 for almost all r ∈ R. Thus, every nonzero derivation d of k[X] is a finite sum of p-homogeneous derivations of the form d_[r]. If r is the maximal real number such that d_[r] 6= 0 then the derivation d_[r] is called the main p-component of d.

The following proposition, which we may find in [2] (Principle II), plays an impor-

Proposition 2.2. The main p-component of a locally nilpotent derivation of k[X] is locally nilpotent. 

3 Jacobian derivations

If w₁, . . . , w_n ∈ k[X], then we denote by J(w₁, . . . , w_n) the Jacobian of w₁, . . . , w_n, i. e.,

J (w₁, . . . , w_n) = det [∂w_i/∂x_j] .

The polynomials w₁, . . . , w_nare algebraically independent over k iff J (w₁, . . . , w_n) 6= 0.

Assume now that w₁, . . . , w_n−1 ∈ k[X] and denote by ∆_{w1,...,wn−1} the mapping from k[X] to k[X] defined by

∆w1,...,wn−1(f ) = J (w1, . . . , wn−1, f ),

for all f ∈ k[X]. It is a derivation of k[X]. Such a derivation is said to be a Jacobian derivation of k[X]. The derivation ∆_{w1,...,wn−1} is nonzero if and only if the polynomials w1, . . . , wn−1 are algebraically independent over k.

Let 0 6= p = (p₁, . . . , p_n) ∈ Rⁿ. The partial derivative ∂/∂x_i (for any i = 1, . . . , n) is a p-homogeneous derivation of k[X] of degree −p_i. This implies that if w₁, . . . , w_n∈ k[X] are p-forms of degrees s1, . . . , sn, respectively, then the Jacobian J (w1, . . . , wn) is a p-form of degree Pn

i=1(s_i − p_i). As a consequence of this fact we get the following two propositions.

Proposition 3.1. If w₁, . . . , w_n−1 ∈ k[X] are p-forms of degrees s₁, . . . , s_n−1, respec- tively, then the derivation ∆w1,...,wn−1 is p-homogeneous of degree Pn−1

i=1 si −Pn i=1pi.



Proposition 3.2. Let ∆ = ∆_{w1,...,wn−1}, where w₁, . . . , w_n−1 ∈ k[X]. Let r =Pn−1

i=1 deg_p(w_i) −Pn i=1p_i.

Consider the p-homogeneous derivation ∆_[r] defined in Section 2. Then we have:

(1) ∆_[r] = ∆w^∗₁,...,w^∗_n−1;

(2) if ∆_[r] 6= 0, then D_[r] is the main p-component of ∆. 

In the next proposition the derivation ∆_{w1,...,wn−1} appears in the case when the polynomials w1, . . . , wn−1 are monomials. Note that monomials X^α1, . . . , X^αs (where α₁, . . . , α_s∈ Ω) are algebraically independent over k if and only if α₁, . . . , α_sare linearly independent over Q.

Proposition 3.3 ([4]). Let α₁, . . . , α_n−1 be linearly independent over Q elements from Ω and consider the derivation

∆ = ∆_Xα1,...,X^αn−1.

If ∆ is locally nilpotent, then there exists i ∈ {1, . . . , n} such that all the monomials X^α1, . . . , X^αn−1 do not admit the variable x_i.

D. See [4] Lemma (page 84). 

Let us end this section with the following

Proposition 3.4. Let g, f₁, . . . , f_n−1 ∈ k[X]. Assume that:

(1) f₁, . . . , f_n−1 are algebraically independent over k, (2) g 6∈ k, and

(3) g, f₁, . . . , f_n−1 are algebraically dependent over k.

Then there exists i ∈ {1, . . . , n−1} such that the polynomials f₁, . . . , f_i−1, g, f_i+1, . . . , f_n−1 are algebraically independent over k.

Proof (Makar - Limanov [8] in the proof of Lemma 3). Denote by M the sequence (f₁, f₂, . . . , f_n−1, g) and let F ∈ k[y₁, . . . , y_n] r k be a polynomial of the minimal degree such that F (M ) = 0. It follows from (1) that the variable y_n appears in F . Moreover, it follows from (2) that a variable y_i with i ∈ {1, . . . , n − 1} also appears in F . We may assume (up to renumbering) that y₁ appears in F . Let a = (∂F/∂y₁)(M ) and b = (∂F/∂y_n)(M ). Then a 6= 0, b 6= 0 and we have:

0 = ∆_{0,f2,...,fn−1}

= ∆_{F (M ),f2,...,fn−1}

= a∆_{f1,f2,...,fn−1}+ b∆_{g1,f2,...,fn−1}.

The derivation ∆_{f1,f2,...,fn−1} is nonzero (since f₁, . . . , f_n−1 are algebraically indepen- dent), hence the derivation ∆_{g1,f2,...,fn−1} is also nonzero, and hence the polynomials g1, f2, . . . , fn−1 are algebraically independent. 

4 Results of Makar-Limanov

Theorem 4.1 ([9]). Let d : k[X] → k[X] be a nonzero locally nilpotent derivation and let f1, . . . , fn−1 be polynomials belonging to k[X]^d. Then

ad = b∆_{f1,...,fn−1},

for some a, b ∈ k[X]^dwith b 6= 0. Moreover, if f₁, . . . , f_n−1are algebraically independent then a 6= 0.

D. See the proof of Lemma 8 (page 22) in [9]. 

Colorary 4.2. Let d be a nonzero locally nilpotent derivation of k[X]. If f₁, . . . , f_n−1 ∈ k[X]^d, then the derivation ∆_{f1,...,fn−1} is locally nilpotent.

D. Put ∆ = ∆_{f1,...,fn−1}. If ∆ = 0 then of course ∆ i locally nilpotent. Assume that ∆ 6= 0. Then f₁, . . . , f_n−1 are algebraically independent and so, by Theorem 4.1, ad = b∆ for some nonzero a, b ∈ k[X]^d. The derivation ad is locally nilpotent. Hence b∆ is locally nilpotent and it is well know (see for example [12]) that then ∆ is also locally nilpotent. 

The partial case of the above corollary we may find in [8] or [6].

5 Proof of Theorem 1.1

Let d be a nonzero locally nilpotent derivation of k[X] and let g ∈ k[X]^d. The case

”g ∈ k” is obvious. Assume that g 6∈ k and let α ∈ Ω be a vertex of N_g.

It is well know (see for example [12]) that tr.deg_kk[X]^d = n − 1. Therefore there exist algebraically independent polynomials f₁, . . . , f_n−1 belonging to k[X]^d. Using Proposition 3.4 (and a renumbering of f₁, . . . , f_n−1) we may assume that the polyno- mials g, f₂, . . . , f_n−1 are algebraically independent over k.

Consider now a hyperplane in Rⁿ intersecting the polytope N_g in the point α and such that α is a unique common point. Let 0 6= p = (p₁, . . . , p_n) ∈ Rⁿ be a direction vector of this hyperplane. We may assume that the point p is primary (see Section 2) and that the p-form g^∗ is associate with X^α. Then every p-form of k[X] is a monomial.

Look now on Proposition 2.1. There exist polynomials Q₁, . . . , Q_n−1 belonging to A = k[g, f₂, . . . , f_n−1] such that Q₁ = g and the p-forms Q^∗₁, . . . , Q^∗_n−1 are algebraically independent over k. Consider the derivations

∆ = ∆_{Q1,...,Qn−1} and D = ∆_Q^∗

1,...,Q^∗_n−1.

The derivation D is nonzero (because Q^∗₁, . . . , Q^∗_n−1 are algebraically independent) and hence, by Proposition 3.2, D is the main p-component of ∆.

Since A ⊆ k[X]^d and Q₁, . . . , Q_n−1 ∈ A, the derivation ∆ is locally nilpotent (by Corollary 4.2) and so, by Proposition 2.2, the derivation D is also locally nilpotent.

Now Proposition 3.3 implies that the monomials Q^∗₁, . . . , Q^∗_n−1 do not admit a variable x_i, for some i ∈ {1, . . . , }. This means, in particular, that the vertex α = (α₁, . . . , α_n) is such that αi = 0 for some i. This completes the proof of Theorem 1.1. 

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