BECKER’S CONJECTURE ON MAHLER FUNCTIONS.

In 1994, Becker conjectured that if F(z) is a k-regular power series, then there exists a k-regular rational function R(z) such that F(z)/R(z) satisfies a Mahler-type functional equation with polynomial coefficients where the initial coefficient satisfies a0(z) = 1. In this paper, we prove Becker’s conjecture in the best-possible form; we show that the rational function R(z) can be taken to be a polynomial zγQ(z) for some explicit nonnegative integer γ and such that 1/Q(z) is k-regular. [ABSTRACT FROM AUTHOR]