Algebraic power series and their automatic complexity modulo prime powers

TL;DR

This paper improves bounds on the automaton size for algebraic sequences modulo prime powers by introducing a new numeration system and embedding sequences as diagonals of rational functions, enhancing understanding of their automatic complexity.

Contribution

It provides a significantly tighter bound on automaton size for algebraic sequences modulo prime powers and introduces a novel numeration system for representing automaton states.

Findings

Automaton size bound improved to p^{α^3 h d}

States are represented in a new numeration system

Bounds extended to diagonals of multivariate rational functions

Abstract

Christol and, independently, Denef and Lipshitz showed that an algebraic sequence of $p$-adic integers (or integers) is $p$-automatic when reduced modulo $p^\alpha$. Previously, the best known bound on the minimal automaton size for such a sequence was doubly exponential in $\alpha$. Under mild conditions, we improve this bound to the order of $p^{\alpha^3 h d}$, where $h$ and $d$ are the height and degree of the minimal annihilating polynomial modulo $p$. We achieve this bound by showing that all states in the automaton are naturally represented in a new numeration system. This significantly restricts the set of possible states. Since our approach embeds algebraic sequences as diagonals of rational functions, we also obtain bounds more generally for diagonals of multivariate rational functions.