[Paper Review] Elimination via saturation

Mohamed Barakat, Markus Lange‐Hegermann+1 more|arXiv (Cornell University)|Jul 4, 2017

Polynomial and algebraic computation5 references3 citations

TL;DR

This paper presents a saturation-based algorithm for elimination in polynomial rings that avoids block-elimination orders by using homogenization and ideal saturation. It shows that the elimination ideal $I \cap B$ can be computed as $I^h \cap B$, where $I^h$ is the homogenization of $I$ via saturation with respect to the homogenizing variable $x_0$, enabling elimination using any monomial order, including degree-reverse-lexicographic order.

ABSTRACT

This short paper presents saturation-based algorithms for homogenization and elimination. This algorithm can compute elimination ideals by using syzygies and ideal membership test, hence it works with any} monomial order, in particular without the use of block-elimination orders. The used saturation is a translation of the geometric fact that the projective closure of an affine scheme has no components in the hyperplane at infinity.

Motivation & Objective

• To answer whether saturation can be used to perform elimination, reversing the usual direction of ideal quotient and saturation computations.
• To investigate whether constructive elimination theory is a special case of constructive module theory, particularly whether non-block orders like degree-reverse-lex can be used for elimination.
• To provide a method for computing elimination ideals $I \cap B$ without relying on block-elimination monomial orders.
• To establish a geometric-algebraic correspondence between projective closure and saturation, translating the fact that $\overline{X}$ has no components in the hyperplane at infinity into algebraic saturation.
• To generalize the method to submodules of free modules over polynomial rings.

Proposed method

• Homogenize the ideal $I \subset R = B[x_1,\dots,x_n]$ using a new variable $x_0$ to form $I^h \subset S = B[x_0,\dots,x_n]$.
• Compute $I^h$ as the saturation $\langle g^h \mid g \in G \rangle : x_0^\infty$, where $G$ is a generating set of $I$.
• Use iterated ideal quotients via syzygy computation to perform the saturation, which is feasible for rings with effective coset representatives.
• Obtain the elimination ideal $I \cap B$ by evaluating $x_0, \dots, x_n$ to zero in a generating set of $I^h$.
• Generalize the method to submodules of $R^\ell$ by homogenizing each generator to make them $S$-homogeneous and saturating the resulting module.
• Leverage the algebraic translation of geometric fact: the projective closure $\overline{X}$ of an affine scheme $X$ has no components in the hyperplane at infinity, which corresponds to $I^h \cap B = I \cap B$.

Experimental results

Research questions

• RQ1 Can saturation be used as a computational tool for elimination, rather than only for ideal quotients or saturations?
• RQ2 Is elimination theory a special case of constructive module theory, allowing the use of non-block monomial orders such as degree-reverse-lexicographic order?
• RQ3 Can the homogenization of an ideal be computed via saturation without using block-elimination Gröbner bases?
• RQ4 Does the equality $I \cap B = I^h \cap B$ hold over arbitrary commutative base rings $B$, without restrictions?
• RQ5 Can the method be extended from ideals to submodules of finite rank free modules?

Key findings

• The elimination ideal $I \cap B$ is equal to $I^h \cap B$, where $I^h$ is the homogenization of $I$ with respect to $x_0$, providing a geometric-algebraic foundation for the method.
• The homogenized ideal $I^h$ can be computed as $\langle g^h \mid g \in G \rangle : x_0^\infty$, where $G$ is a generating set of $I$, enabling computation without block-elimination orders.
• The saturation $J : x_0^\infty$ for $J = \langle g^h \mid g \in G \rangle$ can be computed via iterated ideal quotients using syzygies, which is effective when $B$ has effective coset representatives.
• Evaluating $x_0, \dots, x_n$ to zero in a generating set of $I^h$ yields a generating set for $I \cap B$, completing the elimination process.
• The method generalizes to submodules of $R^\ell$, where homogenization is applied component-wise and saturation is performed in $S^\ell$, with elimination achieved by evaluation to zero.
• The approach is valid over any commutative unital base ring $B$, and the proofs are given in full generality without restricting to fields or Noetherian rings.