Related papers: Algebraic independence for values of integral curv…

We deal with the distributions of holomorphic curves and integral points off divisors. We will simultaneouly prove an optimal dimension estimate from above of a subvariety W off a divisor D which contains a Zariski dense entire holomorphic…

We prove some new degeneracy results for integral points and entire curves on surfaces; in particular, we provide the first example, to our knowledge, of a simply connected smooth variety whose sets of integral points are never…

Let X be a set definable in a sharply o-minimal structure. We consider the problem of counting the number of points where X intersects algebraic varieties V over Q of dimension k < codim X, as a function of T := deg(V) + h(V), where h(V) is…

The purpose of this paper is two-fold. We first prove a series of results, concerned with the notion of Zariski multiplicity, mainly for non-singular algebraic curves. These results are required in the paper "A Theory of Branches for…

We introduce an algebraicity criteria. It has the following form: under certain conditions, an analytic subvariety of some algebriac variety over a global field $K$, if it contains many $K$-points, then it is algebraic over $K.$ This gives…

Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. Our analytic curves identify pairs of circles…

We show a higher order integrability theorem for distributions generated by a family of vector fields under a horizontal regularity assumption on their coefficients. We use as chart a class of almost exponential maps which we discuss in…

For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the…

This paper is concerned with algebraic geometry over complete discretely valued fields $K$ of equicharacteristic zero. Several results are given including: the canonical projection $K^{n} \times K\mathbb{P}^{m} \longrightarrow K^{n}$ and…

Contrary to previous approaches bringing together algebraic geometry and signatures of paths, we introduce a Zariski topology on the space of paths itself, and study path varieties consisting of all paths whose iterated-integrals signature…

Consider some non-zero complex numbers $a_i, b_i, c_i, d_i$ with $1 \leq i \leq n$ and the associated classical Lotka-Volterra systems \[ \begin{cases} x' = a_i xy + b_i y \newline y' = c_i xy + d_i y \text{ .} \end{cases} \] We show that…

The classic Schneider-Lang theorem in transcendence theory asserts that there are only finitely many points at which algebraically independent complex meromorphic functions of finite order of growth can simultaneously take values in a…

There has been a long-standing question about whether being perfectoid for an algebra is local in the analytic topology. We provide affirmative answers for the algebras (e.g., over $\overline{\mathbb{Z}_p}$) whose spectra are inverse limits…

We study the growth of degrees in many autonomous and non-autonomous lattice equations defined by quad rules with corner boundary values, some of which are known to be integrable by other characterisations. Subject to an enabling…

We study entire holomorphic curves in the algebraic torus, and show that they can be characterized by the ``growth rate'' of their derivatives.

We show that for a variety which admits a quasi-finite period map, finiteness (resp.~non-Zariski-density) of $S$-integral points implies finiteness (resp.~non-Zariski-density) of points over all $\mathbb{Z}$-finitely generated integral…

Consider a field $k$ of characteristic $0$, not necessarily algebraically closed, and a fixed algebraic curve $f=0$ defined by a tame polynomial $f\in k[x,y]$ with only quasi-homogeneous singularities. We prove that the space of holomorphic…

We give a generalization of the classical Bombieri--Schneider--Lang criterion in transcendence theory. We give a local notion of $LG$--germ, which is similar to the notion of $E$-- function and Gevrey condition, and which generalize (and…

In this note, we are interested in the Jacobian Conjecture. Following the results of L.M.~Dru$\dot{\rm z}$kowski, we consider some vector fields depending on a certain \'etale polynomial map. From results of semialgebraic geometry with the…

The algebraic degeneracy of holomorphic curves in a semi-Abelian variety omitting a divisor is proved (Lang's conjecture generalized to semi-Abelian varieties) by making use of the {\it jet-projection method} and the logarithmic Wronskian…