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math.AC

math.AC: explore 4 source-linked works published from 2026 to 2026, with original documents and citations.

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Sources: arxiv. Collection updated 2026-09-15. Counts describe this index, not the complete source archives.

arXiv2026

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

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.

math.AC↗

arXiv2026

The Alexander-Hirschowitz theorem for neurovarieties

We study the dimension and identifiability of neurovarieties associated to polynomial neural networks. We give an independent geometric proof that the linear bounds $d_i\geq 2n_i-1$ on the activation degrees imply non defectiveness for any number of outputs, a dimension statement previously obtained from finite identifiability. The proof is based on a direct analysis of the differential of the parameterization. We also investigate secant and Grassmann-secant obstructions outside this range and prove global identifiability for multi-output architectures under the same degree bounds.

math.AG↗

arXiv2026

Edge codes constructed from unicyclic graphs

Jaramillo-Velez recently introduced edge codes, a new class of toric evaluation codes constructed from the edges of a (hyper)graph $\mathcal{H}$. In the case that $\mathcal{H}$ is a tree, Jaramillo-Velez computed both the minimum distance and the weight distribution of the associated code. In this paper, we study edge codes associated to unicyclic graphs. Our most striking result is that computing the parameters of these codes is subtle in the case that the induced cycle has an even length because these values will depend on certain conditions regarding the length of the cycle and the size of the base field.

math.CO↗

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↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Degree bounds and synchronization in Gröbner basis computations for affine semi-regular systems	2026-09-01	2404.03530	arxiv
The Alexander-Hirschowitz theorem for neurovarieties	2026-09-01	2511.19703	arxiv
Edge codes constructed from unicyclic graphs	2026-08-31	2608.30407	arxiv
Learning Fast Monomial Orders for Gröbner Basis Computations	2026-08-28	2602.02972	arxiv

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