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Section: New Results

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, whose trailing coefficient (i.e., of order 0) is 1. In [2], Frédéric Chyzak and Philippe Dumas, together with Jason P. Bell (University of Waterloo, Canada) and Michael Coons (University of Newcastle, Australia) have proved Becker’s conjecture in the best-possible form: they have shown that the rational function R(z) can be taken to be a polynomial zγQ(z) for some explicit non-negative integer γ and such that 1/Q(z) is k-regular. The article was published this year.