Related papers: On some rational generating series occuring in ari…
We prove a closed formula for the generating function $\mathsf Z_d(t)$ of the motives $[\mathrm{Hilb}^d(\mathbb A^n)_0] \in K_0(\mathrm{Var}_{\mathbb C})$ of punctual Hilbert schemes, summing over $n$, for fixed $d>0$. The result is an…
We introduce and study algebraic dynamical systems generated by triangular systems of rational functions. We obtain several results about the degree growth and linear independence of iterates as well as about possible lengths of…
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.
The sequence of the primes $p$ for which a variety over $\mathbb{Q}$ has no $p$-adic point plays a fundamental role in arithmetic geometry. This sequence is deterministic, however, we prove that if we choose a typical variety from a family…
We prove the conjecture of Grosse-Kunstleve et al. that coordination sequences of periodic structures in n-dimensional Euclidean space are rational. This has been recently proven by Nakamura et al.; however, our proof is a straightforward…
This paper introduces and studies a notion of \emph{algorithmic randomness} for subgroups of rationals. Given a randomly generated additive subgroup $(G,+)$ of rationals, two main questions are addressed: first, what are the model-theoretic…
We revisit the classical two-dimensional McKay correspondence in two respects: The first one, which is the main point of this work, is that we take into account of the multiplicative structure given by the orbifold product; second, instead…
We provide a formal, simple and intuitive theory of rational decision making including sequential decisions that affect the environment. The theory has a geometric flavor, which makes the arguments easy to visualize and understand. Our…
Motivated by problems arising in the relative trace formula and arithmetic invariant theory we prove the existence of rational points on orbits arising from certain infinitesimal symmetric spaces. As an application, we prove analogous…
We survey our recent work on an extension of the theory of motivic integration, called arithmetic motivic integration. We developed this theory to understand how p-adic integrals of a very general type depend on p.
The arithmetic motivic Poincar\'e series of a variety $V$ defined over a field of characteristic zero, is an invariant of singularities which was introduced by Denef and Loeser by analogy with the Serre-Oesterl\'e series in arithmetic…
It was shown that in a group of bijections of an infinite set some families of subsets, related to the cardinality of some eigenspaces, are generating. Besides, we derived a criterion for generating by sets of this kind.
Sequences are often conveniently encoded in the form of a generating function depending on a formal variable. This note presents two observations that allow one to draw conclusions about the generated sequence from the generating function.…
We show a possibility to apply certain philosophical concepts to the analysis of concrete mathematical structures. Such application gives a clear justification of topological and geometric properties of considered mathematical objects.
Let $F$ be a field of characteristic zero admitting a biquadratic field extension. We give an example of a torus $G$ over $F$ whose classifying stack $BG$ is stably rational and such that $\{BG\}\{G\}\neq 1$ in the Grothendieck ring of…
We develop a theory of modulus triples, for future motivic applications.
Our goal is to prove that the Leray spectral sequence associated to a map of algebraic varieties is motivic in the following sense: If the singular cohomology groups of the category of quasiprojective varieties defined over a subfield of C…
We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the…
Various methods have been used to construct rational points and rational curves on rationally connected algebraic varieties. We survey recent advances in two of them, the descent and the fibration method, in a number-theoretical context…
A period is a complex number arising as the integral of a rational function with algebraic number coefficients over a rationally-defined region. Although periods are typically transcendental numbers, there is a conjectural Galois theory of…