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