Automatic sequences and curves over finite fields

Bridy, Andrew

doi:10.2140/ant.2017.11.685

Cited by 8 publications

(21 citation statements)

References 18 publications

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“…Another new input we shall use is a globalization argument allowing us to compare section operators at 0 and at f (t). This argument is formalized throught Frobenius operators and is closely related to the Cartier operator used in a beautiful geometric proof of Christol's theorem due to Deligne [11] and Speyer [18], and further studied by Bridy [5].…”

Section: Effective Version Of Christol's Theoremmentioning

confidence: 99%

Fast coefficient computation for algebraic power series in positive characteristic

et al. 2019

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We revisit Christol's theorem on algebraic power series in positive characteristic and propose yet another proof for it. This new proof combines several ingredients and advantages of existing proofs, which make it very wellsuited for algorithmic purposes. We apply the construction used in the new proof to the design of a new efficient algorithm for computing the N th coefficient of a given algebraic power series over a perfect field of characteristic p. It has several nice features: it is more general, more natural and more efficient than previous algorithms. Not only the arithmetic complexity of the new algorithm is linear in log N and quasi-linear in p, but its dependency with respect to the degree of the input is much smaller than in the previously best algorithm. Moreover, when the ground field is finite, the new approach yields an even faster algorithm, whose bit complexity is linear in log N and quasi-linear in √ p.

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Section: Effective Version Of Christol's Theoremmentioning

confidence: 99%

Fast coefficient computation for algebraic power series in positive characteristic

et al. 2019

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“…See e.g. the recent proof due to Bridy[21] (following Speyer[22]), where the author also gave a sharp estimate of the size of DFAO that generates algebraic ξ ∈ F q [[t]] by relating the operatorsρ i on F q [[t]]and Cartier operators on differentials Ω K/Fq on the algebraic curve defined by ξ over F q , and by using Riemann-Roch theorem for function fields over finite fields.3.2 Arithmetic analogueOur analogue of Christol's theorem is arithmetic in the sense that the polynomial ring F q [t] is replaced with the ring O K of integers of a number field K. In this variant, the ring F q [[t]] of formal power series is replaced with the ring W OK (OK ) of Witt vectors (cf. §2) over the Dedekind domain O K with coefficients in algebraic integers OK .…”

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confidence: 99%

Semi-galois categories II: An arithmetic analogue of Christol's theorem

Uramoto

2018

Journal of Algebra

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In connection with our previous work on semi-galois categories [1,2], this paper proves an arithmetic analogue of Christol's theorem concerning an automata-theoretic characterization of when a formal power series ξ = ξnt n ∈ Fq[[t]] over finite field Fq is algebraic over the polynomial ring Fq [t]. There are by now several variants of Christol's theorem, all of which are concerned with rings of positive characteristic. This paper provides an arithmetic (or F1-) variant of Christol's theorem in the sense that it replaces the polynomial ring Fq[t] with the ring OK of integers of a number field K and the ring Fq[[t]] of formal power series with the ring of Witt vectors. We also study some related problems.1 Some background on Eilenberg theory and its relationship to semi-galois categories will be briefly summarized in §5.2 for the neccessity of our discussion. For more comprehensive texts on Eilenberg theory and its historical aspect, the reader is referred to e.g. [19] and [17]; also, for the axiomatization of Eilenberg theory based on semi-galois categories, see §1, §5 and §7 in [2].

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“…Combining a bound of [2] on the state complexity in terms of the expansion complexity and a bound of [16] on the state complexity in terms of the correlation measure of order 2, we can also estimate the expansion complexity in terms of the correlation measure of order 2.…”

Section: Final Remarksmentioning

confidence: 99%

On the Maximum Order Complexity of the Thue-Morse and Rudin-Shapiro Sequence

Sun

Winterhof

2019

Uniform Distribution Theory

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Expansion complexity and maximum order complexity are both finer measures of pseudorandomness than the linear complexity which is the most prominent quality measure for cryptographic sequences. The expected value of the Nth maximum order complexity is of order of magnitude log N whereas it is easy to find families of sequences with Nth expansion complexity exponential in log N. This might lead to the conjecture that the maximum order complexity is a finer measure than the expansion complexity. However, in this paper we provide two examples, the Thue-Morse sequence and the Rudin-Shapiro sequence with very small expansion complexity but very large maximum order complexity. More precisely, we prove explicit formulas for their N th maximum order complexity which are both of the largest possible order of magnitude N. We present the result on the Rudin-Shapiro sequence in a more general form as a formula for the maximum order complexity of certain pattern sequences.

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Automatic sequences and curves over finite fields

Cited by 8 publications

References 18 publications

Fast coefficient computation for algebraic power series in positive characteristic

Fast coefficient computation for algebraic power series in positive characteristic

Semi-galois categories II: An arithmetic analogue of Christol's theorem

On the Maximum Order Complexity of the Thue-Morse and Rudin-Shapiro Sequence

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