Bézout bounds and rational points on quasianalytic graphs

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 $\mathrm{\Gamma }(f)$ 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 $\mathrm{\Gamma }(f)$ 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 $\mathrm{\Gamma }(f)$ contains few rational points: the number of rational points of height at most $H$ is bounded by a polynomial in $\log H$.