Matrix-F5 algorithms over finite-precision complete discrete valuation fields

Let (f(1,) . . . ,f(s)) is an element of Q(p) [X-1,X- . . . ,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 Grobner 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, Grobner bases can be computed stably has many applications. Firstly, the mapping sending (f(1,) . . . , f(s)) to the reduced Grobner 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. (C) 2016 Elsevier Ltd. All rights reserved.