Height gaps for coefficients of D-finite power series

Khoa D. Nguyen (University of Calgary, Canada)

A power series $f(x_1,\ldots,x_m)\in \mathbb{C}[[x_1,\ldots,x_m]]$ is said to be D-finite if all the partial derivatives of $f$ span a finite dimensional vector space over the field $\mathbb{C}(x_1,\ldots,x_m)$. For the univariate series $f(x)=\sum a_nx^n$, this is equivalent to the condition that the sequence $(a_n)$ is P-recursive meaning a non-trivial linear recurrence relation of the form:
$$P_d(n)a_{n+d}+\cdots+P_0(n)a_n=0$$ where the $P_i$'s are polynomials. In this talk, we consider D-finite power series with algebraic coefficients and discuss the growth of the Weil height of these coefficients. This is from a joint work with Jason Bell and Umberto Zannier in 2019 and a more recent work in June 2022.