Tristan Vaccon

Tracking p-adic precision

We present a new method to propagate

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

p

p-Adic Stability In Linear Algebra

-adic precision in computations, which also applies to other ultrametric fields. We illustrate it with many examples and give a toy application to the stable computation of the SOMOS 4 sequence.

On p-Adic Differential Equations with Separation of Variables

Let (<i>f</i>₁,..., <i>f</i>_<i>s</i>) ∈ Q_<i>p</i> [<i>X</i>₁,..., <i>X</i>_<i>n</i>]^<i>s</i> be a sequence of homogeneous polynomials with <i>p</i>-adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since A_<i>p</i> 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 <i>w</i>. 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 ⟨<i>f</i>₁,..., <i>f</i>_<i>i</i>⟩ are weakly-<i>w</i>-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.

Matrix-F5 Algorithms and Tropical Gröbner Bases Computation

Using the differential precision methods developed previously by the same authors, we study the p-adic stability of standard operations on matrices and vector spaces. We demonstrate that lattice-based methods surpass naive methods in many applications, such as matrix multiplication and sums and intersections of subspaces. We also analyze determinants, characteristic polynomials and LU factorization using these differential methods. We supplement our observations with numerical experiments.

A Tropical F5 Algorithm

Several algorithms in computer algebra involve the computation of a power series solution of a given ordinary differential equation. Over finite fields, the problem is often lifted in an approximate

Characteristic Polynomials of p-adic Matrices

p

Matrix-F5 algorithms and tropical Gröbner bases computation

-adic setting to be well-posed. This raises precision concerns: how much precision do we need on the input to compute the output accurately? In the case of ordinary differential equations with separation of variables, we make use of the recent technique of differential precision to obtain optimal bounds on the stability of the Newton iteration. The results apply, for example, to algorithms for manipulating algebraic numbers over finite fields, for computing isogenies between elliptic curves or for deterministically finding roots of polynomials in finite fields. The new bounds lead to significant speedups in practice.

Division and Slope Factorization of p-Adic Polynomials

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gröbner bases taking into account the valuation of K. Because of the use of the valuation, this theory is promising for stable computations over polynomial rings over a p-adic fields. We design a strategy to compute such tropical Gröbner bases by adapting the Matrix-F5 algorithm. Two variants of the Matrix-F5 algorithm, depending on how the Macaulay matrices are built, are available to tropical computation with respective modifications. The former is more numerically stable while the latter is faster. Our study is performed both over any exact field with valuation and some inexact fields like Qp or Fq [[t]]. In the latter case, we track the loss in precision, and show that the numerical stability can compare very favorably to the case of classical Gröbner bases when the valuation is non-trivial. Numerical examples are provided.

ZpL: a p-adic Precision Package

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gröbner bases taking into account the valuation of K. While generalizing the classical theory of Gröbner bases, it is not clear how modern algorithms for computing Gröbner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated F5 Algorithm of Faugère. In this article, we prove that, for homogeneous ideals, it can be adapted to the tropical case. We prove termination and correctness. Because of the use of the valuation, the theory of tropical Gröbner bases is promising for stable computations over polynomial rings over a p-adic field. We provide numerical examples to illustrate time-complexity and p-adic stability of this tropical F5 algorithm.