arXiv subjects

Yuta Kambe

Publications and source records attributed to Yuta Kambe.

7 recordsLinked to original sources

arXiv2025

Geometric Generality of Transformer-Based Gr\"obner Basis Computation

The intersection of deep learning and symbolic mathematics has seen rapid progress in recent years, exemplified by the work of Lample and Charton. They demonstrated that effective training of machine learning models for solving mathematical problems critically depends on high-quality, domain-specific datasets. In this paper, we address the computation of Gr\"obner basis using Transformers. While a dataset generation method tailored to Transformer-based Gr\"obner basis computation has previously been proposed, it lacked theoretical guarantees regarding the generality or quality of the generated datasets. In this work, we prove that datasets generated by the previously proposed algorithm are sufficiently general, enabling one to ensure that Transformers can learn a sufficiently diverse range of Gr\"obner bases. Moreover, we propose an extended and generalized algorithm to systematically construct datasets of ideal generators, further enhancing the training effectiveness of Transformer. Our results provide a rigorous geometric foundation for Transformers to address a mathematical problem, which is an answer to Lample and Charton's idea of training on diverse or representative inputs.

cs.LG↗

arXiv2023

Learning to Compute Gr\"obner Bases

Solving a polynomial system, or computing an associated Gr\"obner basis, has been a fundamental task in computational algebra. However, it is also known for its notorious doubly exponential time complexity in the number of variables in the worst case. This paper is the first to address the learning of Gr\"obner basis computation with Transformers. The training requires many pairs of a polynomial system and the associated Gr\"obner basis, raising two novel algebraic problems: random generation of Gr\"obner bases and transforming them into non-Gr\"obner ones, termed as backward Gr\"obner problem. We resolve these problems with 0-dimensional radical ideals, the ideals appearing in various applications. Further, we propose a hybrid input embedding to handle coefficient tokens with continuity bias and avoid the growth of the vocabulary set. The experiments show that our dataset generation method is a few orders of magnitude faster than a naive approach, overcoming a crucial challenge in learning to compute Gr\"obner bases, and Gr\"obner computation is learnable in a particular class.

math.AC↗

arXiv2023

Analysis of computing Gröbner bases and Gröbner degenerations via theory of signatures

The signatures of polynomials were originally introduced by Faugère for the efficient computation of Gröbner bases [Fau02], and redefined by Arri-Perry [AP11] as the standard monomials modulo the module of syzygies. Since it is difficult to determine signatures, Vaccon-Yokoyama [VY17] introduced an alternative object called guessed signatures. In this paper, we consider a module $\mathrm{Gobs}(F)$ for a tuple of polynomials $F$ to analyse computation of Gröbner bases via theory of signatures. This is the residue module $\mathrm{ini}_{\prec}(\mathrm{Syz}(\mathrm{LM}(F)))/\mathrm{ini}_{\prec}(\mathrm{Syz}(F))$ defined by the initial modules of the syzygy modules with respect to the Schreyer order. We first show that $F$ is a Gröbner basis if and only if $\mathrm{Gobs}(F)$ is the zero module. Then we show that any homogeneous Gröbner basis with respect to a graded term order satisfying a common condition must contain the remainder of a reduction of an S-polynomial. We give computational examples of transitions of minimal free resolutions of $\mathrm{Gobs}(F)$ in a signature based algorithm. Finally, we show a connection between the module $\mathrm{Gobs}(F)$ and Gröbner degenerations.

math.AC↗

arXiv2020

Construction of the moduli space of reduced Groebner bases

For a given monomial ideal $J \subset k[x_1, \ldots, x_n]$ and a given monomial order $\prec$, the moduli functor of all reduced Gröbner bases with respect to $\prec$ whose initial ideal is $J$ is determined. In some cases, such a functor is representable by an affine scheme of finite type over $k$, and a locally closed subfunctor of a Hilbert scheme. The moduli space is called the Gröbner basis scheme, the Gröbner strata and so on if it exists. This paper introduces an alternative procedure for explicitly constructing a defining ideal of the Gröbner basis scheme and its Zariski tangent spaces by studying combinatorics on the standard set associated to $J$. That is a generalization of Robbiano and Lederer's technique. We also see that we can make an implementation of that. Moreover, as a generalization of Robbiano's result, we show that if the Gröbner basis scheme for $\prec$ and $J$ defined over the rational numbers $\mathbb{Q}$ is nonsingular at the $\mathbb{Q}$-rational point corresponding to $J$, then the Gröbner basis scheme for $\prec$ and $J$ defined over any commutative ring $k$ is isomorphic to an affine space over $k$.

math.AG↗

arXiv2020

The Gröbner fan of the Hilbert scheme

We give a notion of "combinatorial proximity" among strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. We show that this notion guarantees "geometric proximity" of the corresponding points in the Hilbert scheme. We define a graph whose vertices correspond to strongly stable ideals and whose edges correspond to pairs of adjacent ideals. Every term order induces an orientation of the edges of the graph. This directed graph describes the behavior of the points of the Hilbert scheme under Gröbner degenerations with respect to the given term order. Then, we introduce a polyhedral fan that we call Gröbner fan of the Hilbert scheme. Each cone of maximal dimension corresponds to a different directed graph induced by a term order. This fan encodes several properties of the Hilbert scheme. We use these tools to present a new proof of the connectedness of the Hilbert scheme. Finally, we improve the technique introduced in the paper "Double-generic initial ideal and Hilbert scheme" by Bertone, Cioffi and Roggero to give a lower bound on the number of irreducible components of the Hilbert scheme.

math.AG↗

arXiv2019

Gröbner scheme in the Hilbert scheme and complete intersection monomial ideals

Let $k$ be a commutative ring and $S=k[x_0, \ldots, x_n]$ be a polynomial ring over $k$ with a monomial order. For any monomial ideal $J$, there exists an affine $k$-scheme of finite type, called Gröbner scheme, which parameterizes all homogeneous reduced Gröbner bases in $S$ whose initial ideal is $J$. Here we functorially show that the Gröbner scheme is a locally closed subscheme of the Hilbert scheme if $J$ is a saturated ideal. In the process, we also show that the Gröbner scheme consists of complete intersections if $J$ defines a complete intersection.

math.AG↗

arXiv2019

Computable Białynicki-Birula decomposition of the Hilbert scheme

We call the scheme parameterizing homogeneous ideals with fixed initial ideal the Gröbner scheme. We introduce a Białynicki-Birula decomposition of the Hilbert scheme $\mathrm{Hilb}^{P}_n$ for any Hilbert polynomial $P$ such that the cells are the Gröbner schemes in set-theoretically. Then we obtain a computable homology formula for smooth Hilbert schemes. As a corollary of our argument, we show that the Gröbner scheme for a monomial ideal defining a smooth point in the Hilbert scheme is smooth.

math.AG↗