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Elimination by Substitution

Martin Kreuzer, Lorenzo Robbiano

Abstract

Let $K$ be a field and $P=K[x_1,\dots,x_n]$. The technique of elimination by substitution is based on discovering a coherently $Z=(z_1,\dots,z_s)$-separating tuple of polynomials $(f_1,\dots,f_s)$ in an ideal $I$, i.e., on finding polynomials such that $f_i = z_i - h_i$ with $h_i \in K[X \setminus Z]$. Here we elaborate on this technique in the case when $P$ is non-negatively graded. The existence of a coherently $Z$-separating tuple is reduced to solving several $P_0$-module membership problems. Best separable re-embeddings, i.e., isomorphisms $P/I \longrightarrow K[X \setminus Z] / (I \cap K[X \setminus Z])$ with maximal $\#Z$, are found degree-by-degree. They turn out to yield optimal re-embeddings in the positively graded case. Viewing $P_0 \longrightarrow P/I$ as a fibration over an affine space, we show that its fibers allow optimal $Z$-separating re-embeddings, and we provide a criterion for a fiber to be isomorphic to an affine space. In the last section we introduce a new technique based on the solution of a unimodular matrix problem which enables us to construct automorphisms of $P$ such that additional $Z$-separating re-embeddings are possible. One of the main outcomes is an algorithm which allows us to explicitly compute a homogeneous isomorphism between $P/I$ and a non-negatively graded polynomial ring if $P/I$ is regular.

Elimination by Substitution

Abstract

Let be a field and . The technique of elimination by substitution is based on discovering a coherently -separating tuple of polynomials in an ideal , i.e., on finding polynomials such that with . Here we elaborate on this technique in the case when is non-negatively graded. The existence of a coherently -separating tuple is reduced to solving several -module membership problems. Best separable re-embeddings, i.e., isomorphisms with maximal , are found degree-by-degree. They turn out to yield optimal re-embeddings in the positively graded case. Viewing as a fibration over an affine space, we show that its fibers allow optimal -separating re-embeddings, and we provide a criterion for a fiber to be isomorphic to an affine space. In the last section we introduce a new technique based on the solution of a unimodular matrix problem which enables us to construct automorphisms of such that additional -separating re-embeddings are possible. One of the main outcomes is an algorithm which allows us to explicitly compute a homogeneous isomorphism between and a non-negatively graded polynomial ring if is regular.
Paper Structure (6 sections, 11 theorems, 26 equations, 8 algorithms)

This paper contains 6 sections, 11 theorems, 26 equations, 8 algorithms.

Table of Contents

  1. Introduction
  2. Coherently Separating Tuples and Elimination by Substitution
  3. Re-Embeddings of Non-Negatively Graded Ideals
  4. Best Separating Re-Embeddings
  5. Optimal Re-Embeddings and Freeness of the Fibers
  6. Re-Embeddings Using the Unimodular Matrix Problem

Key Result

Proposition 2.4

(Computing Elimination By Substitution) Let $K$ be a field, let $P=K[x_1,\dots,x_n]$, let $I=\langle G\rangle$ be an ideal in $P$ generated by a tuple of polynomials $G=\langle g_1,\dots,g_r)$, let $Z=(z_1,\dots,z_s)$ be a tuple of distinct indeterminates in $X$, and let $F=(f_1,\dots,f_s)$ be a coh

Theorems & Definitions (54)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Example 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Definition 2.8
  • ...and 44 more