Structure of lexicographic Groebner bases in three variables of ideals of dimension zero

Let I be a zero-dimensional ideal of a polynomial ring R[x, y, z] over a Noetherian domain R. The lexicographic order ≺:=≺ lex(x,y,z) , for which x ≺ y ≺ z, is put on the monomials of k[x, y, z] Given a polynomial p ∈ k[x, y, z], the leading monomial of p, denoted lm ≺ (p) is the largest monomial for ≺ occurring in p. The coefficient in R in front of lm ≺ (p) is called the leading coefficient of p, denoted lc ≺ (p). It might also be convenient to define the leading term of p denoted lt ≺ (p) equal to lc ≺ (p)lm ≺ (p).

The ideal of leading terms of I is the ideal of R[x, y, z] generated by the leading terms of elements of I; it is equal to lt ≺ (I) . Since R is Noetherian, there is a finite set of generators of this ideal. A Gröbner basis of I is a finite set of elements in I, g 1 , . . . , g s such that lt ≺ (g 1 ), . . . , lt ≺ (g s ) = lt ≺ (I) .

In our case, we will take R = k a field. Note that then lt ≺ (I) is equal to lm ≺ (I) . This last ideal being a monomial ideal, it admits a minimal basis of monomials m 1 , . . . , m s ; Then a Gröbner basis g 1 , . . . , g s is minimal if lm ≺ (g i ) = m i for all i. It is monic if lc ≺ (g i ) = 1 for all i.

From now on, the monomial order will always be assumed to be lex(x, y, z) and th symbol ≺ will be omitted in lm ≺ , lc ≺ and lt ≺ .

Furthermore, let lm 1 (p) and lm 2 (p) be the monomials such that lt(p) = lc 1 (p)lm

Moreover, we make the following assumption: Assumption: The ideal I will be supposed zero-dimensional, or, equivalently the k-algebra k[x, y, z]/I is supposed finite. We are given a minimal and monic Gröbner basis G := {g 1 , . . . , g s } of I, indexed in a way that lm(g 1 ) ≺ lm(g 2 ) ≺ • • • ≺ lm(g s ).

We recall some basic facts about the Gröbner basis G: * Supported by the GCOE program “Math-for-Industry” of Kyûshû university

• g 1 ∈ k[x] and lm(g s ) = z ds for some d s ∈ N ⋆ (we say that lm(g s ) is pure power of z).

• Moreover, there exists 1 < ℓ(2) < s such that: lm(g ℓ( 2) ) = y d ℓ(2) is a pure power of y and such that

• Elimination property: the set of polynomials g 1 , . . . , g ℓ( 2) is a minimal lexicographic Gröbner basis of the zero-dimensional ideal

In 1985, Lazard in [5] proves the following.

Theorem 1 (D. Lazard) Let J ⊂ k[x, y] be a zero-dimensional ideal, and f 1 , . . . , f r a minimal lexicographic Gröbner basis of I for x ≺ lex(x,y) y. Then:

It follows easily a factorization property of the polynomials in such a Gröbner basis [5, Theorem 1 (i)]. However, the formulation above is more compact and handy, and is equivalent. The main result of this paper is the following analogue in the case of 3 variables:

. . , g s } and ℓ(2) be defined as above. Then, for all 1 ≤ j ≤ i ≤ s such that the variable z appears in the monomials lm(g i ) and lm(g j ) with the same exponent, holds:

and if I is radical:

Furthermore, in the later case, for all i > ℓ(2), g i ∈ lc 2 (g i ), g 1 .

The proof will occupy the next section. There is one corollary to this theorem in the context of “stability of Gröbner bases under specialization”, which generalizes the theorem of Gianni-Kalkbrener [2,3], and improves the theorem of Becker [1] (but holds only with 3 variables).

Corollary 1 Let us assume I radical. Let α be a root of g 1 , φ : k[x, y, z] → k[y, z], x → α, and g = g 1 a polynomial among the Gröbner basis. Then, either φ(g) = g(α, x, z) = 0, or φ(lc 1 (g)) = 0. This implies that: lt(φ(g)) = φ(lt(g)), and in particular, that φ(G) is a Gröbner basis.

Proof: By Theorem 2, we can write g = lc 1 (g)A with

Gianni-Kalkbrener’s result [2,3] concerns the easier case where all the variables but the largest one for ≺ are specialized. Gianni-Kalkbrener. The map φ is therein φ

Becker [1] has generalized partly this result to the case of a map φ that specializes the t lowest variables for ≺. Taking t = 1, this covers the case of Corollary 1, but is weaker: it does also say that φ(G) remains a Gröbner basis, while assuming that for g ∈ G, φ(lt(g)) may be a term with a monomial strictly smaller for ≺ than the monomial in the term lt(φ(g)) (see the definition of the integer r ′ during the proof of Prop. 1 page 4 of [1]. With the notations on the same page of [1] we see r ′ < r; Corollary 1 above implies r = r ′ ). It can not be said that:

Concerning previous works, let us mention that Kalkbrener [4] has expanded Becker’s result to the more general elimination monomial orders. Still, staying in the purely lexicographic case, it does not enhance the theorem of Becker.

The main ingredient of the proof consists in generalizing two lemmas of Lazard. These refers to Lemma 2, and Lemma 3 of [5]. We shall explain that a weaker form holds with a larger number of variables. The version of interest here concerns the case of 3 variables. It is nonetheless easy to produce a version with an arbitrary number of variables. Let us first introduce some notations for exponents: Notation 2 Let f ∈ k[x, y, z] non zero, with leading monomial lm(f ) = x a y b z

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