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An elementary proof of Bridy's theorem

Eric Rowland, Manon Stipulanti, Reem Yassawi

TL;DR

The paper provides an elementary proof of Bridy's bound on the size of the minimal $q$-automaton for $q$-automatic sequences arising from algebraic power series over $\mathbb{F}_q$. It achieves this by embedding algebraic sequences as diagonals of rational functions and analyzing the orbit of a Cartier-operator-based linear map $\lambda_{0,0}$ on a fixed finite-dimensional border-structured polynomial space, reducing the problem to univariate border emulations and period arguments. The main result bounds the kernel size by $|\ker_q(a(n))| \le q^{h d} + q^{(h-1)(d-1)} \mathcal{L}(h,d,d) + \lfloor\log_q h\rfloor + \lfloor\log_q \max(h,d)\rfloor + 3$, yielding Bridy’s asymptotic $(1+o(1)) q^{h d}$ as parameters grow. The article also provides numerical evidence supporting sharpness, develops a robust framework for diagonals of rational functions, and notes connections to recent multilinear generalizations, while offering conjectural structure results for univariate orbit fixed points that may guide future refinements.

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.

An elementary proof of Bridy's theorem

TL;DR

The paper provides an elementary proof of Bridy's bound on the size of the minimal -automaton for -automatic sequences arising from algebraic power series over . It achieves this by embedding algebraic sequences as diagonals of rational functions and analyzing the orbit of a Cartier-operator-based linear map on a fixed finite-dimensional border-structured polynomial space, reducing the problem to univariate border emulations and period arguments. The main result bounds the kernel size by , yielding Bridy’s asymptotic as parameters grow. The article also provides numerical evidence supporting sharpness, develops a robust framework for diagonals of rational functions, and notes connections to recent multilinear generalizations, while offering conjectural structure results for univariate orbit fixed points that may guide future refinements.

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.
Paper Structure (8 sections, 25 theorems, 99 equations, 2 figures, 2 tables)

This paper contains 8 sections, 25 theorems, 99 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Numeric evidence for sharpness
  3. The vector space of possible states
  4. Structure of the linear transformation $\lambda_{0, 0}$
  5. The linear structure of $\lambda_{0,0}$
  6. Orbit size of a univariate polynomial under $\lambda_0$
  7. Orbit size under $\lambda_{0, 0}$
  8. Subspaces of univariate polynomials

Key Result

Theorem 1

Let $F = \sum_{n \geq 0} a(n) x^n \in \mathbb{F}_q\llbracket x \rrbracket \setminus \{0\}$ be the Furstenberg series associated with a polynomial $P \in \mathbb{F}_q[x, y]$ of height $h$ and degree $d$. Then the minimal $q$-automaton that generates $a(n)_{n\geq 0}$ has size at most

Figures (2)

  • Figure 1: Partition of the basis of $V$ into seven sets, which generate the subspaces $\left\langle x^0 y^{d - 1} \right\rangle$, $V_\textnormal{t}^\circ$, $\left\langle x^h y^{d - 1} \right\rangle$, $V_\ell^\circ$, $V^\circ$, $V_\textnormal{r}^\circ$, and $\left\langle x^0 y^0 \right\rangle$.
  • Figure 2: Number of polynomials (vertical axis) with degree $d = 2$ that produce unminimized automata with a given size (horizontal axis). The top six plots are for $q = 2$ and vary $h \in \{1, 2, \dots, 6\}$. In the bottom four, $q = 3$ and $h \in \{1, 2, 3, 4\}$.

Theorems & Definitions (59)

  • Definition
  • Theorem 1
  • Theorem 2
  • Example 3
  • Definition
  • Proposition 4
  • Theorem 5: Furstenberg
  • Remark 6
  • Example 7
  • Proposition 8
  • ...and 49 more