Tohoku Mathematical Journal
2017
March
SECOND SERIES VOL. 69, NO. 1

Tohoku Math. J.
69 (2017), 55-65

Title ON THE REDUCTION MODULO $p$ OF MAHLER EQUATIONS

Author Julien Roques

(Received November 27, 2014, revised May 13, 2015)
Abstract. The guiding thread of the present work is the following result, in the vain of Grothendieck's conjecture for differential equations : if the reduction modulo almost all prime $p$ of a given linear Mahler equation with coefficients in $\mathbb{Q}(z)$ has a full set of algebraic solutions, then this equation has a full set of rational solutions. The proof of this result, given at the very end of the paper, relies on intermediate results of independent interest about Mahler equations in characteristic zero as well as in positive characteristic.

Mathematics Subject Classification. Primary 39A06; Secondary 12H10, 65Q20.
Key words and phrases. Mahler equations.

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