On the uniform flatness of polynomial graphs

Abstract

This paper grew out of an effort to understand the so-called uniform flatness, an important concept in the Bargmann--Fock interpolation theory developed by Varolin et al. We show that the graph of every polynomial in one complex variable is a uniformly flat algebraic curve in ${\mathbb{C}}^2$, with vanishing upper density with respect to the Gaussian weight. Combined with a result of Pingali and Varolin, this implies that such graphs are interpolating for the Bargmann--Fock space on ${\mathbb{C}}^2$.

9 pages; completed in July 2025; published in BLMS in March 2026