arXiv subjects

Yuval Salant

Publications and source records attributed to Yuval Salant.

3 recordsLinked to original sources

arXiv2026

Establishing the Polynomial Wolff Axioms for $δ$-Separated $δ$-Tubes With #o-minimality

We establish the full version of a conjecture of Guth and Zahl, giving a lower bound for the volume of a semialgebraic set that has a large intersection with a collection of $δ$-separated $δ$-tubes. Our proof uses o-minimal methods to simplify the proof of Katz and Rogers, who proved the conjecture up to a small factor. We also establish that the constants depend polynomially on the complexity of the semialgebraic set, and more generally in the #o-minimal setting.

math.CA↗

arXiv2026

Sharp zero estimates for trajectories of polynomial vector fields

If $f$ is a tuple of functions satisfying an algebraic ODE and $P\in{\mathbb C}(x)[f]$, it is common in applications to transcendental number theory to consider upper bounds for the order of zero of $P(x,f)$ at a given point in terms of $\operatorname{deg}_x P,\operatorname{deg}_f P$. Nesterenko introduced a condition known as the D-property, which holds in many applications, and proved essentially optimal bounds under this condition. We prove a global form of this result, where the field $\mathbb C(x)$ is replaced by a number field $K$ and $\operatorname{deg}_x P$ is replaced by the logarithmic height $\operatorname{h}(P)$. Under an analogous global D-property, we prove essentially optimal bounds for the number of zeroes, counted with multiplicities, in a fixed compact set. Applications of this result to point-counting theorems are developed in a separate joint paper with Hirata-Kohno and Kawashima.

math.NT↗

arXiv2026

Counting Theorems for Algebraic Relations

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 log-height of V. In particular, we conjecture that after removing a suitable "algebraic part", this number grows polynomially in T -- a generalization of Wilkie's conjecture. We show that this full conjecture implies some open problems in algebraic independence theory. We also formulate a weaker conjecture stating that all intersections above are contained in a poly(T) amount of balls of radius e^{-T}. We then consider the case where X (subset of C^n) is a (compact piece of a) trajectory of a polynomial differential equation satisfying a variant of Nesterenko's D-property. Our main theorem is a proof of the weakened conjecture for such curves when k < sqrt(n) - 1.

math.NT↗