×

Function fields in positive characteristic: expansions and Cobham’s theorem. (English) Zbl 1151.11060

In this paper basically the authors prove the following two theorems:
Theorem 1: Let \(k\) and \(l\) be multiplicatively independent positive integers. A function \(h:\mathbb Q\to\Delta\) is both \(k\)- and \(l\)-automatic if and only if there exist integers \(a\) and \(b\) with \(a>0\) such that:
(1) the sequence \(\{h((n-b)/a)\}_{n\in\mathbb N}\) is eventually periodic;
(2) \(h((x-b)/a)=0\) for \(x\in\mathbb Q\setminus\mathbb N\).
Theorem 2: Let \(p_1\) and \(p_2\) be distinct primes and \(q_1\) and \(q_2\) be powers of \(p_1\) and \(p_2\), respectively. Let \((r_{\alpha})_{\alpha\in\mathbb Q}\) be sequence with well-ordered support and with values lying in a finite set \(A\) with cardinality at most \(\min\{q_1,q_2\}\). Let \(i_1\) and \(i_2\) be injections from \(A\) into \(\mathbb F_{q_1}\) and \(\mathbb F_{q_2}\) respectively. Then the generalized power series \(f(t)=\sum_{\alpha\in\mathbb Q}i_1(r_{\alpha})t^{\alpha}\in \mathbb F_{q_1}((t^Q))\) and \(g(t)=\sum_{\alpha\in\mathbb Q}i_2(r_{\alpha})t^{\alpha}\in \mathbb F_{q_2}((t^Q))\) are both algebraic (respectively over \(\mathbb F_{q_1}(t)\) and \(\mathbb F_{q_2}(t)\)) if and only if there exists a positive integer \(n\) such that \(f(t^n)\) and \(g(t^n)\) are both rational functions.

MSC:

11R58 Arithmetic theory of algebraic function fields
13F25 Formal power series rings
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Allouche, J.-P.; Shallit, J., Automatic Sequences: Theory, Applications, Generalizations (2003), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 1086.11015
[2] Chevalley, C., Introduction to the Theory of Algebraic Functions of One Variable, Math. Surveys, vol. VI (1951), Amer. Math. Soc. · Zbl 0045.32301
[3] Christol, G., Ensembles presques périodiques \(k\)-reconnaissables, Theoret. Comput. Sci., 9, 141-145 (1979) · Zbl 0402.68044
[4] Christol, G.; Kamae, T.; Mendès France, M.; Rauzy, G., Suites algébriques, automates et substitutions, Bull. Soc. Math. France, 108, 401-419 (1980) · Zbl 0472.10035
[5] Cobham, A., On the base-dependence of sets of numbers recognizable by finite automata, Math. Systems Theory, 3, 186-192 (1969) · Zbl 0179.02501
[6] Everest, G.; van der Poorten, A.; Shparlinski, I.; Ward, T., Recurrence Sequences, Math. Surv. Monogr., vol. 104 (2003), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1033.11006
[7] Hahn, H., Über die nichtarchimedische Größensysteme, Gesammelte Abhandlungen, vol. I (1995), Springer-Verlag, (1907), reprinted in:
[8] Kaplansky, I., Maximal fields with valuations, Duke Math. J., 9, 303-321 (1942) · Zbl 0063.03135
[9] Kedlaya, K., The algebraic closure of the power series field in positive characteristic, Proc. Amer. Math. Soc., 129, 3461-3470 (2001) · Zbl 1012.12007
[10] Kedlaya, K., Finite automata and algebraic extensions of function fields, J. Théor. Nombres Bordeaux, 18, 379-420 (2006) · Zbl 1161.11317
[11] Mendès France, M., Sur les décimales des nombres algébriques réels, (Sémin. Théor. Nombres, Exp. No. 28. Sémin. Théor. Nombres, Exp. No. 28, Bordeaux, 1979-1980 (1980), Univ. Bordeaux I: Univ. Bordeaux I Talence), 7 pp · Zbl 0458.10007
[12] Serre, J.-P., Local Fields (1979), Springer-Verlag
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.