Point-counting results for sets in real Euclidean space have found remarkable applications
to diophantine geometry, enabling significant progress on the André–Oort and Zilber–Pink …

Abstract Zilber's Exponential Algebraic Closedness conjecture (also known as Zilber's
Nullstellensatz) gives conditions under which a complex algebraic variety should intersect …

We give a construction of quasiminimal fields equipped with pseudo-analytic maps,
generalizing Zilber's pseudo-exponential function. In particular we construct pseudo …

Zero estimates have a classical history in diophantine approximation and transcendence, as
well as more recent applications to counting rational points on analytic sets. We give some …

Inspired by work done for systems of polynomial exponential equations, we study systems of
equations involving the modular $ j $ function. We show general cases in which these …

We address special cases of the analogues of the exponential algebraic closedness
conjecture relative to the exponential maps of semiabelian varieties and to the modular $ j …

Let $(G, X) $ be a Shimura datum, let $\Omega $ be a connected component of $ X $, let
$\Gamma $ be a congruence subgroup of $ G (\mathbb {Q})^{+} $, and consider the quotient …

We prove some unconditional cases of the Existential Closedness problem for the modular j-
function. For this, we show that for any finitely generated field we can find a “convenient” set …

Inspired by the idea of blurring the exponential function, we define blurred variants of the j-
function and its derivatives, where blurring is given by the action of a subgroup of. For a …

We investigate exponential ideals in the exponential polynomial ring over an exponential
field. We study two notions of maximality for exponential ideals, and relate them to …