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Algebraic power series and their automatic complexity modulo prime powers

Eric Rowland, Reem Yassawi

TL;DR

The paper addresses automaton size bounds for p-automatic sequences given by coefficients of algebraic power series modulo prime powers, improving previous doubly exponential bounds to approximately $p^{\alpha^3 h d}$ by introducing a base-$p/Q$ numeration system that represents automaton states as Laurent polynomials. It embeds algebraic sequences as diagonals of rational functions and extends the analysis to diagonals of multivariate rational functions, using a decomposition of the state space into borders and interior plus univariate emulation to bound orbit sizes. The main contributions include (i) a tight asymptotic bound for Furstenberg-type series, (ii) a compatibility framework for automata across moduli, (iii) period-length results for coefficient sequences modulo $p$ and $p^\alpha$ that feed into orbit-size bounds, and (iv) generalization to diagonals of multivariate rational functions with explicit bounds. The results have practical implications for efficient automaton construction and for understanding automaticity phenomena in diagonals of rational functions, offering a pathway to scalable computation in number-theoretic settings and beyond.

Abstract

Christol and, independently, Denef and Lipshitz showed that an algebraic sequence of $p$-adic integers (or integers) is $p$-automatic when reduced modulo $p^α$. Previously, the best known bound on the minimal automaton size for such a sequence was doubly exponential in $α$. Under mild conditions, we improve this bound to the order of $p^{α^3 h d}$, 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.

Algebraic power series and their automatic complexity modulo prime powers

TL;DR

The paper addresses automaton size bounds for p-automatic sequences given by coefficients of algebraic power series modulo prime powers, improving previous doubly exponential bounds to approximately by introducing a base- numeration system that represents automaton states as Laurent polynomials. It embeds algebraic sequences as diagonals of rational functions and extends the analysis to diagonals of multivariate rational functions, using a decomposition of the state space into borders and interior plus univariate emulation to bound orbit sizes. The main contributions include (i) a tight asymptotic bound for Furstenberg-type series, (ii) a compatibility framework for automata across moduli, (iii) period-length results for coefficient sequences modulo and that feed into orbit-size bounds, and (iv) generalization to diagonals of multivariate rational functions with explicit bounds. The results have practical implications for efficient automaton construction and for understanding automaticity phenomena in diagonals of rational functions, offering a pathway to scalable computation in number-theoretic settings and beyond.

Abstract

Christol and, independently, Denef and Lipshitz showed that an algebraic sequence of -adic integers (or integers) is -automatic when reduced modulo . Previously, the best known bound on the minimal automaton size for such a sequence was doubly exponential in . Under mild conditions, we improve this bound to the order of , where and are the height and degree of the minimal annihilating polynomial modulo . 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.
Paper Structure (11 sections, 51 theorems, 215 equations, 1 figure, 2 tables)

This paper contains 11 sections, 51 theorems, 215 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. The module of possible states
  3. A numeration system for the automaton states
  4. Compatibility of automata
  5. First bounds on the size of the automaton
  6. Structure of the linear transformation $\lambda_{0, 0}$
  7. Period lengths of series expansions modulo $p$ and modulo $p^\alpha$
  8. Orbit size of a univariate Laurent polynomial under $\lambda_0$
  9. Orbit size under $\lambda_{0, 0}$
  10. Non-Furstenberg series
  11. Diagonals of rational functions

Key Result

Theorem 1

Let $p$ be a prime, let $\alpha \geq 1$, and let $F = \sum_{n \geq 0} a(n) x^n \in \mathbb{Z}_p\llbracket x \rrbracket \setminus \{0\}$ be the Furstenberg series associated with a polynomial $P \in \mathbb{Z}_p[x, y]$. Let $h \mathrel{\mathop:}= \deg_x(P \bmod p)$ and $d \mathrel{\mathop:}= \deg_y(P as any of $p$, $\alpha$, $h$, or $d$ tends to infinity and the others remain constant.

Figures (1)

  • Figure 1: Nested polygons for $k\in \{0,1,2\}$ containing pairs of exponents $(i, j)$ corresponding to monomials $x^i y^j$ in the basis of $W_k$, with $(p, \alpha, h, d) = (3,3,2,4)$.

Theorems & Definitions (96)

  • Definition
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4: Denef and Lipshitz
  • Definition
  • Proposition 5
  • Lemma 6
  • Remark 7
  • Proposition 8
  • ...and 86 more