### A characterization of the Bernoulli and Euler polynomials

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We study the probability distribution of the location of a particle performing a cyclic random motion in ${\mathbb{R}}^{d}$. The particle can take n possible directions with different velocities and the changes of direction occur at random times. The speed-vectors as well as the support of the distribution form a polyhedron (the first one having constant sides and the other expanding with time t). The distribution of the location of the particle is made up of two components: a singular component (corresponding...

We prove a quantitative version of a result of Furstenberg [20] and Deligne [14] stating that the diagonal of a multivariate algebraic power series with coefficients in a field of positive characteristic is algebraic. As a consequence, we obtain that for every prime $p$ the reduction modulo $p$ of the diagonal of a multivariate algebraic power series $f$ with integer coefficients is an algebraic power series of degree at most ${p}^{A}$ and height at most $A{p}^{A}$, where $A$ is an effective constant that only depends on...