An elementary proof of Bridy's theorem

TL;DR

This paper provides a new proof of Bridy's theorem, linking algebraic power series over finite fields to automata, by embedding sequences as diagonals of rational functions, thus connecting algebraic and automaton representations.

Contribution

It introduces a novel proof technique for Bridy's bound using diagonals of rational functions, offering an alternative to algebraic geometry methods.

Findings

New proof of Bridy's bound established

Sequences embedded as diagonals of rational functions

Enhanced understanding of automaton size for algebraic sequences

Abstract

Christol's theorem states that a power series with coefficients in a finite field is algebraic if and only if its coefficient sequence is automatic. A natural question is how the size of a polynomial describing such a sequence relates to the size of an automaton describing the same sequence. Bridy used tools from algebraic geometry to bound the size of the minimal automaton for a sequence, given its minimal polynomial. We produce a new proof of Bridy's bound by embedding algebraic sequences as diagonals of rational functions.