On the generating series of the degree sequence
Quang-Khai Nguyen
TL;DR
This paper analyzes the generating series for degree sequences of dominant toric self-maps on projective toric varieties. It shows that, under a spectral gap condition on the associated matrix, the circle of convergence is a natural boundary, implying transcendence and non-holonomicity of the generating series; in dimension two the results are sharp and yield a dichotomy: either the series is rational or its circle is a natural boundary, with reductions modulo large primes being transcendental. The methods fuse toric geometry, convex-geometry (mixed volumes), Fourier analysis of piecewise-linear functions, and automata-theoretic results (Christol’s theorem) to deduce both analytic and arithmetic consequences. The work also demonstrates a rigidity phenomenon for monomial surface maps and extends to a discussion of Artin–Mazur-type zeta functions within this toric setting. Overall, it reveals substantial analytic complexity in degree sequences even when dynamical degrees are easily computed, and it links geometric, combinatorial, and number-theoretic aspects of toric dynamics.
Abstract
We study the generating series associated with the degree sequence of a monomial self-map of a projective toric variety. We establish conditions under which this series has its circle of convergence as a natural boundary, and hence is a transcendental, non-holonomic function. In the case of toric surfaces, our results are sharp; moreover, we answer a question of Bell by proving that their reductions modulo $p$ are transcendental for all but finitely many prime numbers $p$.