arXiv subjects
Publications and source records attributed to Riccardo Tosi.
We express a family of basic cellular integrals over moduli spaces of curves explicitly in terms of multiple zeta values, answering a question of Brown. Moreover, we study a priori the weights appearing in these integrals and find a relation that expresses the odd-dimensional integrals in terms of the even-dimensional ones. We also sketch an explanation of this relation in the spirit of Grothendieck's Period Conjecture.
We compute the periods associated with a special class of hyperplane arrangements. In particular, we exhibit a combinatorial condition on the intersection lattice of a hyperplane arrangement that ensures that its associated periods are linear combinations of special values of multiple polylogarithms. Our method generalizes Brown's approach to the periods of moduli spaces of curves of genus zero. We apply this result to the reflection arrangement of the full monomial group, whose periods are shown to be linear combinations of values of multiple polylogarithms at roots of unity.
Let $α_1,α_2$ be non-zero algebraic numbers such that $\frac{\log α_2}{\logα_1}\notin\mathbb{Q}$ and let $β$ be a quadratic irrational number. In this article, we prove that the values of two relatively prime polynomials $P(x,y,z)$ and $Q(x,y,z)$ with integer coefficients are not too small at the point $\left(\frac{\logα_2}{\log α_1},α_1^β, α_2^β\right)$. We also establish a measure of algebraic independence of those numbers among $\frac{\logα_2}{\log α_1}$, $α^β_1$ and $α^β_2$ which are algebraically independent.
We generalize a result by Vasil'ev on the algebraic independence of periods of abelian varieties to the case when some of these periods are replaced by their exponentials. We eventually derive some applications to values of the beta function at rational points.