Related papers: Some cases of Vojta's Conjecture on integral point…
We prove some new instances of a conjecture of Bachoc, Couvreur and Z\'emor that generalizes Freiman's $3k-4$ Theorem to a multiplicative version in a function field setting. As a consequence we find that if $F$ is a rational function field…
The aim of this paper is to derive new results about Jacobi's inversion formulas for modular forms of levels 5 and 6. For this purpose, we use Farkas and Kra's theory of theta functions with rational characteristics.
We provide in this paper an upper bound for the number of rational points on a curve defined over a one variable function field over a finite field. The bound only depends on the curve and the field, but not on the Jacobian variety of the…
We present an improved incremental selection algorithm of the selection algorithm presented in [1] and prove all the selected conjectures.
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 translate Davenport's and Heilbronn's work on a quantitative version of the Oppenheim conjecture for indefinite diagonal quadratic forms in 5 variables into the setting of function fields.
We formulate extensions of Wilking's Jacobi field splitting theorem to uniformly positive sectional curvature and also to positive and nonnegative intermediate Ricci curvatures.
A simple and elementary derivation of values at integer points for the Riemann's zeta and related functions is reported.
In 2012 the first named author conjectured that totally real quartic fields of fundamental discriminant are determined by the isometry class of the integral trace zero form; such conjecture was based on computational evidence and the analog…
The lecture, given at the ICM 2014 in Seoul, explores several problems of analytic number theory in the context of function fields over a finite field, where they can be approached by methods different than those of traditional analytic…
This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits…
The zeta function of a motive over a finite field is multiplicative with respect to the direct sum of motives. It has beautiful analytic properties, as were predicted by the Weil conjectures. There is also a multiplicative zeta function,…
A family of congruences interpolating between those of Wilson and Giuga is constructed. Several elementary results are established, in order to present a possible approach to establishing Giuga's conjecture.
We show for all $1/2 \le \sigma \le 1$ and $t\ge 3$ that $\zeta(\sigma+it)| \le 76.2 t^{4.45 (1-\sigma)^{3/2}}$, where $\zeta$ is the Riemann zeta function. This significantly improves the previous bounds, where $4.45$ is replaced by…
In this article, we consider the set of points for the holding of the equality in a weighted version of Suita conjecture for higher derivatives, and give relations between the set and the integer valued points of a class of harmonic…
In this note we generalize and prove a recent conjecture of Varchenko concerning the number of critical points of a (multivalued) meromorphic function $\phi$ on an algebraic manifold. Under certain conditions, this number turns out to…
We survey Vojta's higher-dimensional generalizations of the $abc$ conjecture and Szpiro's conjecture as well as recent developments that apply them to various problems in arithmetic dynamics. In particular, the "$abcd$ conjecture" implies a…
In the paper, some lower bounds for polygamma functions are refined.
For lambda phi^4 models, the introduction of a large field cutoff improves significantly the accuracy that can be reached with perturbative series but the calculation of the modified coefficients remains a challenging problem. We show that…
Three-dimensional conformal field theories (CFTs) with slightly broken higher spin symmetry provide an interesting laboratory to study general properties of CFTs and their roles in the AdS/CFT correspondence. In this work we compute the…