arXiv subjects

Tristan Vaccon

Publications and source records attributed to Tristan Vaccon.

At least 19 recordsLinked to original sources

arXiv2026

On semidefinite-representable sets over valued fields

Polyhedra and spectrahedra over the real numbers, or more generally their images under linear maps, are respectively the feasible sets of linear and semidefinite programming, and form the family of semidefinite-representable sets. This paper studies analogues of these sets, as well as the associated optimization problems, when the data are taken over a valued field $K$. For $K$-polyhedra and linear programming over $K$ we present an algorithm based on the computation of Smith normal forms. We prove that fundamental properties of semidefinite-representable sets extend to the valued setting. In particular, we exhibit examples of non-polyhedral $K$-spectrahedra, as well as sets that are semidefinite-representable over $K$ but are not $K$-spectrahedra.

math.AG↗

arXiv2025

Gr{ö}bner bases over polytopal affinoid algebras

Polyhedral affinoid algebras have been introduced by Einsiedler, Kapranov and Lind to connect rigid analytic geometry (analytic geometry over non-archimedean fields) and tropical geometry. In this article, we present a theory of Gr{ö}bner bases for polytopal affinoid algebras that extends both Caruso et al.'s theory of Gr{ö}bner bases on Tate algebras and Pauer et al.'s theory of Gr{ö}bner bases on Laurent polynomials. We provide effective algorithms to compute Gr{ö}bner bases for both ideals of Laurent polynomials and ideals in polytopal affinoid algebras. Experiments with a Sagemath implementation are provided.

cs.SC↗

arXiv2025

Geometric Generality of Transformer-Based Gröbner 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 Gröbner basis using Transformers. While a dataset generation method tailored to Transformer-based Gröbner 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 Gröbner 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↗

arXiv2024

Learning to Compute Gröbner Bases

Solving a polynomial system, or computing an associated Gröbner 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 Gröbner basis computation with Transformers. The training requires many pairs of a polynomial system and the associated Gröbner basis, raising two novel algebraic problems: random generation of Gröbner bases and transforming them into non-Gröbner ones, termed as backward Gröbner 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 Gröbner bases, and Gröbner computation is learnable in a particular class.

math.AC↗

arXiv2024

Universal Analytic Gr{ö}bner Bases and Tropical Geometry

A universal analytic Gr{ö}bner basis (UAGB) of an ideal of a Tate algebra is a set containing a local Gr{ö}bner basis for all suitable convergence radii. In a previous article, the authors proved the existence of finite UAGB's for polynomial ideals, leaving open the question of how to compute them. In this paper, we provide an algorithm computing a UAGB for a given polynomial ideal, by traversing the Gr{ö}bner fan of the ideal. As an application, it offers a new point of view on algorithms for computing tropical varieties of homogeneous polynomial ideals, which typically rely on lifting the computations to an algebra of power series. Motivated by effective computations in tropical analytic geometry, we also examine local bases for more general convergence conditions, constraining the radii to a convex polyhedron. In this setting, we provide an algorithm to compute local Gr{ö}bner bases and discuss obstacles towards proving the existence of finite UAGBs. CCS CONCEPTS $\bullet$ Computing methodologies $\rightarrow$ Algebraic algorithms.

cs.SC↗

arXiv2023

Pourchet's theorem in action: decomposing univariate nonnegative polynomials as sums of five squares

Pourchet proved in 1971 that every nonnegative univariate polynomial with rational coefficients is a sum of five or fewer squares. Nonetheless, there are no known algorithms for constructing such a decomposition. The sole purpose of the present paper is to present a set of algorithms that decompose a given nonnegative polynomial into a sum of six (five under some unproven conjecture or when allowing weights) squares of polynomials. Moreover, we prove that the binary complexity can be expressed polynomially in terms of classical operations of computer algebra and algorithmic number theory.

cs.SC↗

arXiv2022

On Polynomial Ideals And Overconvergence In Tate Algebras

In this paper, we study ideals spanned by polynomials or overconvergent series in a Tate algebra. With state-of-the-art algorithms for computing Tate Gr{ö}bner bases, even if the input is polynomials, the size of the output grows with the required precision, both in terms of the size of the coefficients and the size of the support of the series. We prove that ideals which are spanned by polynomials admit a Tate Gr{ö}bner basis made of polynomials, and we propose an algorithm, leveraging Mora's weak normal form algorithm, for computing it. As a result, the size of the output of this algorithm grows linearly with the precision. Following the same ideas, we propose an algorithm which computes an overconvergent basis for an ideal spanned by overconvergent series. Finally, we prove the existence of a universal analytic Gr{ö}bner basis for polynomial ideals in Tate algebras, compatible with all convergence radii.

cs.SC↗

arXiv2021

Fast evaluation of some p-adic transcendental functions

We design algorithms for computing values of many p-adic elementary and special functions, including logarithms, exponentials, polylogarithms, and hypergeometric functions. All our algorithms feature a quasi-linear complexity with respect to the target precision and most of them are based on an adaptation to the-adic setting of the binary splitting and bit-burst strategies.

cs.SC↗

arXiv2021

Signature-based algorithms for Gr{ö}bner bases over Tate algebras

Introduced by Tate in [Ta71], Tate algebras play a major role in the context of analytic geometry over the-adics, where they act as a counterpart to the use of polynomial algebras in classical algebraic geometry. In [CVV19] the formalism of Gr{ö}bner bases over Tate algebras has been introduced and effectively implemented. One of the bottleneck in the algorithms was the time spent on reduction , which are significantly costlier than over polynomials. In the present article, we introduce two signature-based Gr{ö}bner bases algorithms for Tate algebras, in order to avoid many reductions. They have been implemented in SageMath. We discuss their superiority based on numerical evidences.

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arXiv2021

On FGLM Algorithms with Tate Algebras

Tate introduced in [Ta71] the notion of Tate algebras to serve, in the context of analytic geometry over the-adics, as a counterpart of polynomial algebras in classical algebraic geometry. In [CVV19, CVV20] the formalism of Gr{ö}bner bases over Tate algebras has been introduced and advanced signature-based algorithms have been proposed. In the present article, we extend the FGLM algorithm of [FGLM93] to Tate algebras. Beyond allowing for fast change of ordering, this strategy has two other important benefits. First, it provides an efficient algorithm for changing the radii of convergence which, in particular, makes effective the bridge between the polynomial setting and the Tate setting and may help in speeding up the computation of Gr{ö}bner basis over Tate algebras. Second, it gives the foundations for designing a fast algorithm for interreduction, which could serve as basic primitive in our previous algorithms and accelerate them significantly.

cs.SC↗

arXiv2020

On FGLM Algorithms with Tropical Gröbner bases

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{ö}bner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gr{ö}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{ö}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{ö}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{ö}bner basis from one weight to one for another weight. Secondly, the FGLM strategy can be applied to go from a tropical Gr{ö}bner basis to a classical Gr{ö}bner basis. We provide tools to chain the stable computation of a tropical Gr{ö}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

On a non-archimedean broyden method

Newton's method is an ubiquitous tool to solve equations, both in the archimedean and non-archimedean settings -- for which it does not really differ. Broyden was the instigator of what is called "quasi-Newton methods". These methods use an iteration step where one does not need to compute a complete Jacobian matrix nor its inverse. We provide an adaptation of Broyden's method in a general non-archimedean setting, compatible with the lack of inner product, and study its Q and R convergence. We prove that our adapted method converges at least Q-linearly and R-superlinearly with R-order $2^{\frac{1}{2m}}$ in dimension m. Numerical data are provided.

cs.SC↗

arXiv2020

Super-linear convergence in the p-adic QR-algorithm

The QR-algorithm is one of the most important algorithms in linear algebra. Its several variants make feasible the computation of the eigenvalues and eigenvectors of a numerical real or complex matrix, even when the dimensions of the matrix are enormous. The first adaptation of the QR-algorithm to local fields was given by the first author in 2019. However, in this version the rate of convergence is only linear and in some cases the decomposition into invariant subspaces is incomplete. We present a refinement of this algorithm with a super-linear convergence rate in many cases.

math.NA↗

arXiv2019

Gr{ö}bner bases over Tate algebras

Tate algebras are fundamental objects in the context of analytic geometry over the p-adics. Roughly speaking, they play the same role as polynomial algebras play in classical algebraic geometry. In the present article, we develop the formalism of Gr{ö}bner bases for Tate algebras. We prove an analogue of the Buchberger criterion in our framework and design a Buchberger-like and a F4-like algorithm for computing Gr{ö}bner bases over Tate algebras. An implementation in SM is also discussed.

math.AG↗

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{ö}bner bases taking into account the valuation of $K$.Because of the use of the valuation, the theory of tropical Gr{ö}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{ö}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{ö}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↗

arXiv2018

ZpL: a p-adic precision package

We present a new package ZpL for the mathematical software system SM. It implements a sharp tracking of precision on p-adic numbers, following the theory of ultrametric precision introduced in [4]. The underlying algorithms are mostly based on automatic dierentiation techniques. We introduce them, study their complexity and discuss our design choices. We illustrate the bene-ts of our package (in comparison with previous implementations) with a large sample of examples coming from linear algebra, com-mutative algebra and dierential equations.

math.NT↗

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{ö}bner bases taking into account the valuation of K. While generalizing the classical theory of Gr{ö}bner bases, it is not clear how modern algorithms for computing Gr{ö}bner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated F5 Algorithm of Faug{è}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{ö}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↗

arXiv2017

Characteristic polynomials of p-adic matrices

We analyze the precision of the characteristic polynomial of an $n\times n$ p-adic matrix A using differential precision methods developed previously. When A is integral with precision O(p^N), we give a criterion (checkable in time O~(n^omega)) for $χ$(A) to have precision exactly O(p^N). We also give a O~(n^3) algorithm for determining the optimal precision when the criterion is not satisfied, and give examples when the precision is larger than O(p^N).

math.NT↗