Related papers: Potentiel et rationalit\'e

We obtain a recursive formula for the number of rational degree $d$ curves in $\mathbb{P}^3$, whose image lies in a $\mathbb{P}^2$, passing through $r$ lines and $s$ points, where $r + 2s = 3d+2$. This can be viewed as a family version of…

In this paper, we explore several threads arising from our recent joint work on arithmetic holonomy bounds, which were originally devised to prove new irrationality results based on the method of Ap\'ery limits. We propose a new method to…

Let X be an irreducible variety and Bir(X) its group of birational transformations. We show that the group structure of Bir(X) determines whether X is rational and whether X is ruled. Additionally, we prove that any Borel subgroup of Bir(X)…

Kolmogorov's axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional…

Bounding the number of rational points of height at most $H$ on irreducible algebraic plane curves of degree $d$ has been an intense topic of investigation since the work by Bombieri and Pila. In this paper we establish optimal dependence…

We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets…

We prove a quantitative theorem for Diophantine approximation by rational points on spheres. Our results are valid for arbitrary unimodular lattices and we further prove 'spiraling' results for the direction of approximates. These results…

We show that the parabola is of strong Khintchine type for convergence, which is the first result of its kind for curves. Moreover, Jarnik type theorems are established in both the simultaneous and the dual settings, without monotonicity on…

We propose a novel constructive framework for approaching the Hodge Conjecture via explicit degenerations. Building on limiting mixed Hodge structures (LMHS), we formulate a criterion under which a rational class of type (p, p) on a smooth…

Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gr\"obner basis…

This paper presents a formalized proof of a discrete form of the Jordan Curve Theorem. It is based on a hypermap model of planar subdivisions, formal specifications and proofs assisted by the Coq system. Fundamental properties are proven by…

Let $n$ be a positive integer. We show that a unit rational space vector whose multiple by $n$ is an integer vector can be extended to a rational orthonormal basis whose all members have the same property.

We study singular rational curves in projective space, deducing conditions on their parametrizations from the value semigroups $\sss$ of their singularities. In particular, we prove that a natural heuristic for the codimension of the space…

Let k be a number field. It is well known that the set of sequences composed by Taylor coefficients of rational functions over k is closed under component-wise operations, and so it can be equipped with a ring structure. A conjecture due to…

These expository notes discuss the arithmetic of rationally connected varieties. Detailed proofs of theorems of Koll\'ar, of Koll\'ar and Szab\'o and of Esnault about rationally connected varieties over finite fields and local fields are…

I provide a systematic construction of points, defined over finite radical extensions, on any Legendre curve over any field of characteristic not equal two. This includes as special case Douglas Ulmer's construction of rational points over…

A method for the analytical evaluation of layer potentials arising in the collocation boundary element method for the Laplace and Helmholtz equation is developed for piecewise flat boundary elements with polynomial shape functions. The…

To the best of our knowledge, a complete characterization of the domains that escape the famous Arrow's impossibility theorem remains an open question. We believe that different ways of proving Arrovian theorems illuminate this problem.…

We discuss some examples that illustrate the countability of the positive rational numbers and related sets. Techniques include radix representations, Godel numbering, the fundamental theorem of arithmetic, continued fractions, Egyptian…

We use Hodge theory to prove a new upper bound on the ranks of Mordell-Weil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and…