Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2404.03530

Degree bounds and synchronization in Gröbner basis computations for affine semi-regular systems

Momonari Kudo·Kazuhiro Yokoyama
View original source ↗Read source document ↗Export metadata ↓

Abstract

Determining the complexity of computing Gröbner bases is an important problem in both theory and practice, and solving degrees provide a central measure of this complexity. We study solving degrees and Gröbner basis computations for affine polynomial systems, with particular emphasis on semi-regular sequences. We first derive two upper bounds for the maximum Gröbner basis degree of the homogenized system. One is based on a regular initial subsequence of the highest-degree homogeneous parts. When these parts form a semi-regular sequence in nondecreasing degree order, the bound involves the $n$ smallest input degrees together with the largest one. The other bound is expressed in terms of the saturation exponent with respect to the homogenizing variable. Both are obtained by bounding the degree from which the Hilbert function of the quotient ring associated with the homogenized system is constant. We then compare the Buchberger-like Gröbner basis computations for an affine system, its homogenization, and its highest-degree homogeneous parts. The first degree fall is characterized by failure of injectivity of multiplication by the homogenizing variable. Before that point, choices of S-pairs and reducers in any computation can be matched in the others, and reduction sequences, remainders, intermediate bases, and leading monomials correspond under specialization. Cryptographic semi-regularity guarantees this correspondence until the step degree first reaches the degree of regularity. At that degree, affine reduction steps that preserve the sugar degree lift to homogeneous ones, yielding upper bounds on the algorithmic solving degree for a computation starting directly from the affine input.

Explore related subjects

math.ACcs.CRcs.SCmath.AG

Keep this discovery

BibTeXRIS

Momonari Kudo, Kazuhiro Yokoyama. 2026-09-01. Degree bounds and synchronization in Gröbner basis computations for affine semi-regular systems. https://arxiv.org/abs/2404.03530

Cite the original work for its findings. Save a collection to share your selection of sources.

Discover connections

cs.CR subjectcs.SC subjectmath.AC subjectmath.AG subject

Connections use source metadata and explicit phrase matches, not verified experimental comparisons.

KEEP EXPLORING

Related papers

arXiv2026

Learning Fast Monomial Orders for Gröbner Basis Computations

The efficiency of Gröbner basis computation, the standard engine for solving systems of polynomial equations, depends on the choice of monomial ordering. Despite a near-continuum of possible monomial orders, most implementations rely on static heuristics such as GrevLex, guided primarily by expert intuition. We address this gap by casting the selection of monomial orderings as a reinforcement learning problem over the space of admissible orderings. Our approach leverages domain-informed reward signals that accurately reflect the computational cost of Gröbner basis computations and admits efficient Monte Carlo estimation. Experiments on benchmark problems from systems biology and computer vision show that the resulting learned policies consistently outperform standard heuristics, yielding substantial reductions in computational cost. Moreover, we find that these policies resist distillation into simple interpretable models, providing empirical evidence that deep reinforcement learning allows the agents to exploit non-linear geometric structure beyond the scope of traditional heuristics.

cs.SC↗
arXiv2026

A Note on Binary Quadratic Systems and their relation to complexity theory

Deciding whether a system of multivariate quadratic equations over $\mathbb F_2$ has a solution is a classical NP-complete problem, and remains so for square systems, with as many equations as variables. The hardness of this problem is one of the cornerstones of nowadays post-quantum cryptography. Let $\MQ_0(n)$ and $\MQ_1(n)$ denote the sets of square quadratic systems in $n$ variables having respectively no solutions and exactly one solution. $\cup_{n\geq 2} \MQ_0(n)$ is a coNP-complete language, while $\cup_{n\geq 2} \MQ_1(n)$ lies in DP. It is known that $\lim_{n\to \infty} |\MQ_1(n)|/|\MQ_0(n)|=1$. Here we prove the explicit finite-$n$ bounds \[ |\MQ_0(n)|<|\MQ_1(n)| \le \left(1+\frac{1}{2^n-1}\right)|\MQ_0(n)|, \] More generally, let $Q_d$ be the space of polynomial functions $(\FF_2)^n\to\mathbb F_2$ of degree at most $d$, and let $α_k$ count square systems in $(Q_d)^n$ having exactly $k$ solutions. Then \[ α_0<α_1 \le \left(1+\frac{1}{2^n-1}\right)α_0\,, \qquad 2\le d\le n \,. \] The proof combines matroid and coding-theoretic methods. We interpret $(\FF_2)^n$ as the ground set of the evaluation matroid of $Q_d$, express $α_0$ and $α_1$ through characteristic polynomials, and use a Whitney-type sign-reversing involution to show that the only terms that can push $α_1-α_0$ below $α_1/2^n$ come from the elements of a matroid port. These are identified with minimal-support words of the Reed--Muller code $\RM(n-d-1,n)=\RM(d,n)^\perp$; the required estimate then follows from the MacWilliams identity, the minimum-distance bound $2^{d+1}$, and the even-weight structure of the code.

cs.IT↗