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- W4300576720 abstract "Let $(f_1,dots, f_s) in mathbb{Q}_p [X_1,dots, X_n]^s$ be a sequence of homogeneous polynomials with $p$-adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since $mathbb{Q}_p$ is not an effective field, classical algorithm does not apply.We provide a definition for an approximate Gr{o}bner 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 $langle f_1,dots,f_i rangle$ 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 lean 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, Gr{o}bner bases can be computed stably has many applications. Firstly, the mapping sending $(f_1,dots,f_s)$ to the reduced Gr{o}bner basis of the ideal they span is differentiable, and its differential can be given explicitly. Secondly, these hypotheses allows to perform lifting on the Grobner bases, from $mathbb{Z}/p^k mathbb{Z}$ to $mathbb{Z}/p^{k+k'} mathbb{Z}$ or $mathbb{Z}.$ 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." @default.
- W4300576720 created "2022-10-03" @default.
- W4300576720 creator A5061019239 @default.
- W4300576720 date "2014-03-20" @default.
- W4300576720 modified "2023-09-26" @default.
- W4300576720 title "Matrix-F5 algorithms over finite-precision complete discrete valuation fields" @default.
- W4300576720 doi "https://doi.org/10.48550/arxiv.1403.5464" @default.
- W4300576720 hasPublicationYear "2014" @default.
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