Related papers: On some rational generating series occuring in ari…

We introduce a generating function associated to the homogeneous generators of a graded algebra that measures how far is this algebra from being finitely generated. For the case of some algebras of Frobenius endomorphisms we describe this…

Let $F(x)=\sum\limits_{n=1}^\infty\tau(n)x^n$ be the generating function for the number $\tau(n)$ of spanning trees in the circulant graphs $C_{n}(s_1,s_2,\ldots,s_k).$ We show that $F(x)$ is a rational function with integer coefficients…

This article shows that under general conditions, p-adic orbital integrals of definable functions are represented by virtual Chow motives. This gives an explicit example of the philosophy of Denef and Loeser, which predicts that all…

Gevrey series are ubiquitous in analysis; any series satisfying some (possibly non-linear) analytic differential equation is Gevrey of some rational order. The present work stems from two observations: 1) the classical Gevrey series, e.g.…

We prove a version of Chatelet's Theorem about Severi-Brauer variety having rational points in the setting of synthetic algebraic geometry. We work over an arbitrary base ring.

Consider a subfield of the field of rational functions in several indeterminates. We present an algorithm that, given a set of generators of such a subfield, finds a simple generating set. We provide an implementation of the algorithm and…

For any positive integer $r$, we construct a smooth complex projective rational surface which has at least $r$ real forms not isomorphic over $\mathbb{R}$.

A triangular field of rational numbers is characterized, with relations to Stirling numbers 2nd, Hyperbolic functions, and centered Binomial distribution. A Generating function is given.

We will show that, in the the geometric Lorenz flow, the set of initial states which give rise to orbits with historic behavior is residual in a trapping region.

We define motivic iterated integrals on the affine line, and give a simple proof of the formula for the coproduct in the Hopf algebra of they make. We show that it encodes the group law in the automorphism group of certain non-commutative…

We discuss some examples of geometrically meaningful rational self-maps of moduli space of curves of low genus and homogeneous forms.

We prove that the Euler-Chow series for ruled surfaces and scrolls is rational by means of an explicit computation.

We introduce a direct image formalism for constructible motivic functions. One deduces a very general version of motivic integration for which a change of variables theorem is proved. These constructions are generalized to the relative…

We show that the intersection of the rational derived series of a one-relator group is rationally perfect and is normally generated by a single element. As a corollary, we characterise precisely when a one-relator group is residually…

This survey paper, to appear in he proceedings of the Miami Winter School ``Geometric Methods in Algebra and Number Theory'', is concerned with extending classical results \`a la Ax-Kochen-Er{\v{s}}ov to $p$-adic integrals in a motivic…

The moduli spaces of trigonal curves are proven to be rational when the genus is divisible by 4.

We introduce the notion of numerical semigroups generated by concatenation of arithmetic sequences and show that this class of numerical semigroups exhibit multiple interesting behaviours.

We develop the theory of motivic integration for formal schemes

Let $X$ be a smooth variety, $E$ a locally free sheaf on $X$. We express the generating function of the motives $[\textrm{Quot}_X(E,n)]$ in terms of the power structure on the Grothendieck ring of varieties. This extends a recent result of…

We consider paths in the plane with $(1,0),$ $(0,1),$ and $(a,b)$-steps that start at the origin, end at height $n,$ and stay to the left of a given non-decreasing right boundary. We show that if the boundary is periodic and has slope at…