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Algebraic functions over a field of positive characteristic and Hadamard products. (English) Zbl 0612.12018

Let \(K\) be a field. \(K[[ X]]\) will denote the ring of formal power series in several commuting variables, \(X=(x_ 1,x_ 2,...,x_ k)\). \(K((X))\) will denote the field of fractions of \(K[[X]]\). An element \(f\in K((X))\) is said to be an algebraic function over K if f is algebraic over the field of rational functions \(K(X)\). If further \(f\in K[[X]]\), then \(f\) is said to be an algebraic series over \(K\).
For the case of one variable G. Christol, T. Kamae, M. Mendes-France, and G. Rauzy [Bull. Soc. Math. Fr. 108, 401-419 (1980; Zbl 0472.10035)] have characterized the algebraic functions over a finite field in terms of automata. We adapt their argument to obtain the corresponding result for the case of several variables and over an arbitrary field of positive characteristic. H. Furstenberg [J. Algebra 7, 271-277 (1967; Zbl 0175.03903)] has shown that if K is a finite field and \(f=\sum _{n\geq 0}a_ nx^ n\), \(g=\sum _{n\geq 0}b_ nx^ n\in K[[x]]\) are algebraic, then the Hadamard product of \(f\) and \(g\), \(f*g=\sum _{n\geq 0}a_ nb_ nx^ n\), is also algebraic. His argument still applies when \(K\) is a perfect field and can be extended to any field of positive characteristic using P. Deligne’s result [Invent. Math. 76, 129-143 (1984; Zbl 0538.13007)].
We apply the above characterization of the algebraic series in several variables to generalize this result and prove that if \(f=\sum _{\iota}a_{\iota}X^{\iota}\), \(g=\sum _{\iota}b_{\iota}X^{\iota}\in K[[X]]\) are algebraic series, then \(f*g=\sum _{\iota}a_{\iota}b_{\iota}X^{\iota}\) is also an algebraic series. Finally we deduce Deligne’s theorem as an easy consequence of our main theorem: If \(f\in K[[X]]\) and \(f=\sum _{\sigma}a_{\sigma}X^{\sigma}\) is an algebraic series in \(X\) over \(K\), then \(I(f)=\sum _{n\geq 0}a_{n.\mathbf{1}}t^ n\) is an algebraic series in \(t\) over \(K\).

MSC:

12E99 General field theory
11R58 Arithmetic theory of algebraic function fields
13F25 Formal power series rings
11T99 Finite fields and commutative rings (number-theoretic aspects)
11J81 Transcendence (general theory)
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