arXiv subjects

Tristan Vaccon

Publications and source records attributed to Tristan Vaccon.

25 records · Page 2Linked to original sources

arXiv2016

On p-adic differential equations with separation of variables

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 $p$-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.

cs.SC↗

arXiv2016

On the p-adic stability of the FGLM algorithm

Nowadays, many strategies to solve polynomial systems use the computation of a Gr{ö}bner basis for the graded reverse lexicographical ordering, followed by a change of ordering algorithm to obtain a Gr{ö}bner basis for the lexicographical ordering. The change of ordering algorithm is crucial for these strategies. We study the p-adic stability of the main change of ordering algorithm, FGLM. We show that FGLM is stable and give explicit upper bound on the loss of precision occuring in its execution. The variant of FGLM designed to pass from the grevlex ordering to a Gr{ö}bner basis in shape position is also stable. Our study relies on the application of Smith Normal Form computations for linear algebra.

cs.SC↗

arXiv2016

GraphPrints: Towards a Graph Analytic Method for Network Anomaly Detection

This paper introduces a novel graph-analytic approach for detecting anomalies in network flow data called GraphPrints. Building on foundational network-mining techniques, our method represents time slices of traffic as a graph, then counts graphlets -- small induced subgraphs that describe local topology. By performing outlier detection on the sequence of graphlet counts, anomalous intervals of traffic are identified, and furthermore, individual IPs experiencing abnormal behavior are singled-out. Initial testing of GraphPrints is performed on real network data with an implanted anomaly. Evaluation shows false positive rates bounded by 2.84% at the time-interval level, and 0.05% at the IP-level with 100% true positive rates at both.

cs.CR↗

arXiv2015

Matrix-F5 algorithms and tropical Gröbner bases computation

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 $\mathbb{Q}\_p$ or $\mathbb{F}\_q \llbracket t \rrbracket.$ 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.

cs.SC↗

arXiv2015

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

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{ö}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{ö}bner bases can be computed stably has many applications. Firstly, the mapping sending $(f\_1,\dots,f\_s)$ to the reduced Gr{ö}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.

cs.SC↗

arXiv2015

p-Adic Stability In Linear Algebra

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.

math.NT↗

arXiv2014

Tracking p-adic precision

We present a new method to propagate $p$-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.

math.NT↗