Related papers: $S$-Unit Equations and the Asymptotic Fermat Conje…
A celebrated analogy between prime factorizations of integers and cycle decompositions of permutations is explored here. Asymptotic formulas characterizing semismooth numbers (possessing at most several large factors) carry over to random…
We give a heuristic argument supporting conjectures of Bhargava on the asymptotics of the number of $S_n$-number fields having bounded discriminant. We then make our arguments rigorous in the case $n=3$ giving a new elementary proof of the…
In this comment we show that the ambiguity of entropic quantities calculated in Physical Review A 83, 062323 (2011) for fermionic fields in the context of Unruh effect is not related to the properties of anticommuting fields, as claimed in…
We obtain upper bounds on the number of finite sets $\mathcal S$ of primes below a given bound for which various $2$ variable $\mathcal S$-unit equations have a solution.
Let $K$ be a number field. Using the modular method, we prove asymptotic results on solutions of the Diophantine equation $x^4-y^2=z^p$ over $K$, assuming some deep but standard conjectures of the Langlands programme when $K$ has at least…
The aim of this note is to show the existence of a correspondance between certain algebraic continued fractions in fields of power series over a finite field and automatic sequences in the same finite field. this connection is illustrated…
We state a conjectural relationship between the fermionic form (G. Hatayama, A. Kuniba, M. Okado, T. Takagi, Y. Yamada qa/9812022) and the Betti numbers of a Grassmannian over a preprojective algebra or, equivalently, of a lagrangian quiver…
We compute the whole asymptotic expansion of the probability that a large uniform labeled graph is connected, and of the probability that a large uniform labeled tournament is irreducible. In both cases, we provide a combinatorial…
In a recent work the authors prove the effective asymptotic Fermat's Last Theorem for the infinite family of fields $\mathbb{Q}(\zeta_{2^{r+2}})^+$ where $r \ge 0$. A crucial step in their proof is the following conjecture of Kraus. Let $K$…
Recent results of Freitas, Kraus, Sengun and Siksek give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over various number fields. In this paper, we prove asymptotic results about the solutions of the Diophantine…
In this paper we continue our earlier investigations into the asymptotic behaviour of infinite systems of coupled differential equations. Under the mild assumption that the so-called characteristic function of our system is completely…
We review some surprising links which have been discovered in the last few years between the theory of certain ordinary differential equations, and particular integrable lattice models and quantum field theories in two dimensions. An…
In order to illustrate a recently derived covariant formalism for computing asymptotic symmetries and asymptotically conserved superpotentials in gauge theories, the case of gravity with minimally coupled scalar fields is considered and the…
We study an asymptotic formula for counting integral points over an equation defined by a non-degenerated indefinite integral ternary quadratic form $f$ representing a non-zero integer $a$ such that $-a\cdot det(f)$ is square over a number…
We survey recent joint work with M. Rapoport and W. Zhang related to the arithmetic Gan-Gross-Prasad conjecture for Shimura varieties attached to unitary groups.
We study polynomial summation over unit circles over finite fields of odd characteristic, obtaining a purely algebraic integration theory without recourse to infinite procedures. There are nonetheless strong parallels to classical…
The improvement of resummation algorithms for divergent perturbative expansions in quantum field theory by asymptotic information about perturbative coefficients is investigated. Various asymptotically optimized resummation prescriptions…
The Fermat numbers have many notable properties, including order universality, coprimality, and definition by a recurrence relation. We use arbitrary elliptic curves and rational points of infinite order to generate sequences that are…
We prove an asymptotic formula for the number of multi-quadratic number fields of bounded discriminant with a power-saving error term. Furthermore, we explicitly calculate the leading coefficient and extend our result to totally real…
In this work we report a new result that appears when one investigates the route that starts from a scalar field theory and ends on a supersymmetric quantum mechanics. The subject has been studied before in several distinct ways and here we…