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

Abstract

Let (f₁,..., f_s) ∈ Q_p [X₁,..., X_n]^s be a sequence of homogeneous polynomials with p-adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since A_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 ⟨f₁,..., 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 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.