PreprintCC BY 4.0arXiv
Bézout bounds and rational points on quasianalytic graphs
Abstract
For functions f : [0,1] → [0,1] in quasianalytic Denjoy-Carleman classes 𝒞M, 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.
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BibTeX
@misc{rainer2026bézout,
title = {{Bézout bounds and rational points on quasianalytic graphs}},
author = {Armin Rainer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21725},
eprint = {2610.21725},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21725},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bézout bounds and rational points on quasianalytic graphs
AU - Armin Rainer
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21725
UR - https://arxiv.org/abs/2610.21725
AB - For functions f : [0,1] → [0,1] in quasianalytic Denjoy-Carleman classes 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.
ER -