Year: 2026
Type: preprint
Abstract: For functions $f : [0,1] \to [0,1]$ in quasianalytic Denjoy-Carleman classes $\mathcal C_M$, we establish explicit bounds for the number of intersection points of the graph $Γ(f)$ with algebraic curves of degree at most $d$. The bounds are given in terms of the Bang degree determined by the weight sequence $M$, the degree $d$, and the norming constant of $Γ(f)$ with respect to the space of real bivariate polynomials of degree at most $d$. We give instances, where the bounds are at most polynomial in $d$ and, consequently, the graph $Γ(f)$ contains few rational points: the number of rational points of height at most $H$ is bounded by a polynomial in $\log H$.
Source: arXiv (Cornell University)
Author: Rainer, Armin
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Austrian Science Fund
OpenAlex ID: W7222477512