#### Vol. 11, No. 3, 2017

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Automatic sequences and curves over finite fields

### Andrew Bridy

Vol. 11 (2017), No. 3, 685–712
##### Abstract

We prove that if $y={\sum }_{n=0}^{\infty }$a$\left(n\right){x}^{n}\in {\mathbb{F}}_{q}\left[\left[x\right]\right]$ is an algebraic power series of degree $d$, height $h$, and genus $g$, then the sequence a is generated by an automaton with at most ${q}^{h+d+g-1}$ states, up to a vanishingly small error term. This is a significant improvement on previously known bounds. Our approach follows an idea of David Speyer to connect automata theory with algebraic geometry by representing the transitions in an automaton as twisted Cartier operators on the differentials of a curve.

##### Keywords
automatic sequences, formal power series, algebraic curves, finite fields
##### Mathematical Subject Classification 2010
Primary: 11B85
Secondary: 11G20, 14H05, 14H25