arXiv subjects

Kazuhiro Yokoyama

Publications and source records attributed to Kazuhiro Yokoyama.

9 recordsLinked to original sources

arXiv2024

Generalization of semi-regular sequences: Maximal Gr\"{o}bner basis degree, variants of genericness, and related conjectures

Nowadays, the notion of semi-regular sequences, originally proposed by Fr\"oberg, becomes very important not only in Mathematics, but also in Information Science, in particular Cryptology. For example, it is highly expected that randomly generated polynomials form a semi-regular sequence, and based on this observation, secure cryptosystems based on polynomial systems can be devised. In this paper, we deal with a semi-regular sequence and its extension, named a generalized cryptographic semi-regular sequence, and give precise analysis on the complexity of computing a Gr\"obner basis of the ideal generated by such a sequence with help of several regularities of the ideal related to Lazard's bound on maximal Gr\"{o}bner basis degree and other bounds. We also study the genericness of the property that a sequence is semi-regular, and its variants related to Fr\"oberg's conjecture. Moreover, we discuss on the genericness of another important property that the initial ideal is weakly reverse lexicographic, related to Moreno-Soc\'{i}as' conjecture, and show some criteria to examine whether both Fr\"oberg's conjecture and Moreno-Soc\'{i}as' one hold at the same time.

math.AC↗

arXiv2024

Comprehensive Systems for Primary Decompositions of Parametric Ideals

We present an effective method for computing parametric primary decomposition via comprehensive Gr\"obner systems. In general, it is very difficult to compute a parametric primary decomposition of a given ideal in the polynomial ring with rational coefficients $\mathbb{Q}[A,X]$ where $A$ is the set of parameters and $X$ is the set of ordinary variables. One cause of the difficulty is related to the irreducibility of the specialized polynomial. Thus, we introduce a new notion of ``feasibility'' on the stability of the structure of the ideal in terms of its primary decomposition, and we give a new algorithm for computing a so-called comprehensive system consisting of pairs $(C, \mathcal{Q})$, where for each parameter value in $C$, the ideal has the stable decomposition $\mathcal{Q}$. We may call this comprehensive system a parametric primary decomposition of the ideal. Also, one can also compute a dense set $\mathcal{O}$ such that $\varphi_\alpha(\mathcal{Q})$ is a primary decomposition for any $\alpha\in C\cap \mathcal{O}$ via irreducible polynomials. In addition, we give several computational examples to examine the effectiveness of our new decomposition.

cs.SC↗

arXiv2024

Degree bounds and synchronization in Gr\"{o}bner basis computations for affine semi-regular systems

Determining the complexity of computing Gr\"obner 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 Gr\"obner basis computations for affine polynomial systems, with particular emphasis on semi-regular sequences. We first derive two upper bounds for the maximum Gr\"obner 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 Gr\"obner 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↗

arXiv2024

On Hilbert-Poincar\'{e} series of affine semi-regular polynomial sequences and related Gr\"{o}bner bases

Gr\"{o}bner bases are nowadays central tools for solving various problems in commutative algebra and algebraic geometry. A typical use of Gr\"{o}bner bases is the multivariate polynomial system solving, which enables us to construct algebraic attacks against post-quantum cryptographic protocols. Therefore, the determination of the complexity of computing Gr\"{o}bner bases is very important both in theory and in practice: One of the most important cases is the case where input polynomials compose an (overdetermined) affine semi-regular sequence. The first part of this paper aims to present a survey on Gr\"{o}bner basis computation and its complexity. In the second part, we shall give an explicit formula on the (truncated) Hilbert-Poincar\'{e} series associated to the homogenization of an affine semi-regular sequence. Based on the formula, we also study (reduced) Gr\"{o}bner bases of the ideals generated by an affine semi-regular sequence and its homogenization. Some of our results are considered to give mathematically rigorous proofs of the correctness of methods for computing Gr\"{o}bner bases of the ideal generated by an affine semi-regular sequence.

cs.SC↗

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↗

arXiv2020

On FGLM Algorithms with Tropical Gr\"obner bases

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gr{\"o}bner bases has proved to provide settings for computations over polynomial rings over a p-adic field that are more stable than that of classical Gr{\"o}bner bases. In this article, we investigate how the FGLM change of ordering algorithm can be adapted to the tropical setting. As the valuations of the polynomial coefficients are taken into account, the classical FGLM algorithm's incremental way, monomo-mial by monomial, to compute the multiplication matrices and the change of basis matrix can not be transposed at all to the tropical setting. We mitigate this issue by developing new linear algebra algorithms and apply them to our new tropical FGLM algorithms. Motivations are twofold. Firstly, to compute tropical varieties, one usually goes through the computation of many tropical Gr{\"o}bner bases defined for varying weights (and then varying term orders). For an ideal of dimension 0, the tropical FGLM algorithm provides an efficient way to go from a tropical Gr{\"o}bner basis from one weight to one for another weight. Secondly, the FGLM strategy can be applied to go from a tropical Gr{\"o}bner basis to a classical Gr{\"o}bner basis. We provide tools to chain the stable computation of a tropical Gr{\"o}bner basis (for weight [0,. .. , 0]) with the p-adic stabilized variants of FGLM of [RV16] to compute a lexicographical or shape position basis. All our algorithms have been implemented into SageMath. We provide numerical examples to illustrate time-complexity. We then illustrate the superiority of our strategy regarding to the stability of p-adic numerical computations.

cs.SC↗

arXiv2020

Effective Localization Using Double Ideal Quotient and Its Implementation

In this paper, we propose a new method for localization of polynomial ideal, which we call "Local Primary Algorithm". For an ideal $I$ and a prime ideal $P$, our method computes a $P$-primary component of $I$ after checking if $P$ is associated with $I$ by using "double ideal quotient" $(I:(I:P))$ and its variants which give us a lot of information about localization of $I$.

math.AC↗

arXiv2018

On Affine Tropical F5 Algorithms

Let $K$ be a field equipped with a valuation. Tropical varieties over $K$ can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of $K$.Because of the use of the valuation, the theory of tropical Gr{\"o}bner bases has proved to provide settings for computations over polynomial rings over a $p$-adic field that are more stable than that of classical Gr{\"o}bner bases.Beforehand, these strategies were only available for homogeneous polynomials. In this article, we extend the F5 strategy to a new definition of tropical Gr{\"o}bner bases in an affine setting.We provide numerical examples to illustrate time-complexity and $p$-adic stability of this tropical F5 algorithm.We also illustrate its merits as a first step before an FGLM algorithm to compute (classical) lex bases over $p$-adics.

cs.SC↗

arXiv2017

A Tropical F5 algorithm

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. While generalizing the classical theory of Gr{\"o}bner bases, it is not clear how modern algorithms for computing Gr{\"o}bner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated F5 Algorithm of Faug{\`e}re. In this article, we prove that, for homogeneous ideals, it can be adapted to the tropical case. We prove termination and correctness. Because of the use of the valuation, the theory of tropical Gr{\"o}b-ner bases is promising for stable computations over polynomial rings over a p-adic field. We provide numerical examples to illustrate time-complexity and p-adic stability of this tropical F5 algorithm.

cs.SC↗