Matrix-F5 algorithms over finite-precision complete discrete valuation fields. (May 2017)

• Record Type: Journal Article
• Authors: Vaccon, Tristan
• Is Part Of: Journal of symbolic computation. Volume 80:Part 2(2017)
• Page Start: 329
• Page End: 350
• Publication Date: 2017-05
• Subjects: F5 algorithm -- Gröbner bases -- Moreno-Socias conjecture -- p-Adic algorithm -- p-Adic precision -- Differential precision; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2016.05.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: Let ( f 1, …, f s ) ∈ Q p [ X 1, …, X n ] s be a sequence of homogeneous polynomials with p -adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since Q p is not an effective field, classical algorithm does not apply. We provide a definition for an approximate Gröbner basis with respect to a monomial order w . We design a strategy to compute such a basis, when precision is enough and under the assumption that the input sequence is regular and the ideals 〈 f 1, …, f i 〉 are weakly- w -ideals. The conjecture of Moreno-Socias states that for the grevlex ordering, such sequences are generic. Two variants of that strategy are available, depending on whether one leans more on precision or time-complexity. For the analysis of these algorithms, we study the loss of precision of the Gauss row-echelon algorithm, and apply it to an adapted Matrix-F5 algorithm. Numerical examples are provided. Moreover, the fact that under such hypotheses, Gröbner bases can be computed stably has many applications. Firstly, the mapping sending ( f 1, …, f s ) to the reduced Gröbner basis of the ideal they span is differentiable, and its differential can be given explicitly. Secondly, these hypotheses allow to perform lifting on the Grobner bases, from Z / p k Z to Z / p k + k ′ Z or Z . Finally, asking for the same hypotheses on the highest-degree homogeneous components of the entry polynomials allows to extend our strategy to the affine case.