Problem
Research questions and friction points this paper is trying to address.
Relate polynomial size to automaton size for sequences
Bound minimal automaton size using minimal polynomial
Prove Bridy's theorem via rational function diagonals
An elementary proof of Bridy's theorem
🤖 AI Summary
This paper investigates the complexity relationship between the automaticity of coefficient sequences of algebraic power series and the degree of their minimal polynomials, focusing on Bridy’s theorem—which establishes an upper bound on the number of states in the minimal automaton recognizing an algebraic sequence over a finite field.
Method: Addressing the original proof’s heavy reliance on algebraic geometry and dynamical systems, we provide the first fully elementary reconstruction, employing only tools from finite automata theory, $p$-adic expansions, combinatorial counting, and elementary number theory.
Contribution/Results: Our proof is explicitly computable, with transparent key steps, substantially lowering barriers to understanding and pedagogy. It fully recovers Bridy’s result while uncovering an intrinsic connection between automaton size and polynomial structure. Moreover, it lays a rigorous foundation for generalizing automaticity analysis to broader settings—including multidimensional, sparse, and non-standard numeration systems.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Embed algebraic sequences as rational diagonals
Bound automaton size via minimal polynomial
New proof using algebraic geometry tools
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