A remarkable sequence of integers

A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and number-theoretical nature.Comment: 20 pages, 5 figure

A Generic Approach to Searching for Jacobians

We consider the problem of finding cryptographically suitable Jacobians. By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, we can search for Jacobians containing a large subgroup of prime order. For a suitable distribution of curves, the complexity is subexponential in genus 2, and O(N^{1/12}) in genus 3. We give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size, improving previous results. Our approach is particularly effective over low-degree extension fields, where in genus 2 we find Jacobians over F_{p^2) and trace zero varieties over F_{p^3} with near-prime orders up to 372 bits in size. For p = 2^{61}-1, the average time to find a group with 244-bit near-prime order is under an hour on a PC.Comment: 22 pages, to appear in Mathematics of Computatio

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

From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules

In 1935 J.G. van der Corput introduced a sequence which has excellent uniform distribution properties modulo 1. This sequence is based on a very simple digital construction scheme with respect to the binary digit expansion. Nowadays the van der Corput sequence, as it was named later, is the prototype of many uniformly distributed sequences, also in the multi-dimensional case. Such sequences are required as sample nodes in quasi-Monte Carlo algorithms, which are deterministic variants of Monte Carlo rules for numerical integration. Since its introduction many people have studied the van der Corput sequence and generalizations thereof. This led to a huge number of results. On the occasion of the 125th birthday of J.G. van der Corput we survey many interesting results on van der Corput sequences and their generalizations. In this way we move from van der Corput's ideas to the most modern constructions of sequences for quasi-Monte Carlo rules, such as, e.g., generalized Halton sequences or Niederreiter's $(t,s)$-sequences