On alternative definition of Lucas atoms and their pp-adic valuations

Lucas atoms are irreducible factors of Lucas polynomials and they were introduced in \cite{ST}. The main aim of the authors was to investigate, from an innovatory point of view, when some combinatorial rational functions are actually polynomials. In this paper, we see that the Lucas atoms can be introduced in a more natural and powerful way than the original definition, providing straightforward proofs for their main properties. Moreover, we fully characterize the $p$-adic valuations of Lucas atoms for any prime $p$, answering to a problem left open in \cite{ST}, where the authors treated only some specific cases for $p \in \{2, 3\}$. Finally, we prove that the sequence of Lucas atoms is not holonomic, contrarily to the Lucas sequence that is a linear recurrent sequence of order two

Extended Rate, more GFUN

We present a software package that guesses formulae for sequences of, for example, rational numbers or rational functions, given the first few terms. We implement an algorithm due to Bernhard Beckermann and George Labahn, together with some enhancements to render our package efficient. Thus we extend and complement Christian Krattenthaler's program Rate, the parts concerned with guessing of Bruno Salvy and Paul Zimmermann's GFUN, the univariate case of Manuel Kauers' Guess.m and Manuel Kauers' and Christoph Koutschan's qGeneratingFunctions.m.Comment: 26 page

Short addition sequences for theta functions

International audienceThe main step in numerical evaluation of classical Sl2 (Z) modular forms and elliptic functions is to compute the sum of the first N nonzero terms in the sparse q-series belonging to the Dedekind eta function or the Jacobi theta constants. We construct short addition sequences to perform this task using N + o(N) multiplications. Our constructions rely on the representability of specific quadratic progressions of integers as sums of smaller numbers of the same kind. For example, we show that every generalised pentagonal number c 5 can be written as c = 2a + b where a, b are smaller generalised pentagonal numbers. We also give a baby-step giant-step algorithm that uses O(N/ log r N) multiplications for any r > 0, beating the lower bound of N multiplications required when computing the terms explicitly. These results lead to speed-ups in practice

Affine Algebraic Geometry

Affine geometry deals with algebro-geometric questions of affine varieties that are treated with methods coming from various areas of mathematics like commutative and non-commutative algebra, algebraic, complex analytic and differential geometry, singularity theory and topology. The conference had several main topics. One of them was the famous Jacobian problem, its connections with the Dixmier conjecture and possible algebraic approaches and reductions. A second main theme were questions on Log algebraic varieties, in particular log algebraic surfaces. Thirdly, results on automorphisms of An played a major role, in particular the solution of the Nagata problem, actions of algebraic groups on An , Hilbert’s 14th problem and locally nilpotent derivations. More generally automorphism groups of affine and non-affine varieties, especially in dimension 2 and 3 were treated, and substantial progress on the cancelation problem and embedding problem was presented