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## Section: New Results

### Becker's conjecture on Mahler functions

In 1994, Becker conjectured that if $F\left(z\right)$ is a $k$-regular power series, then there exists a $k$-regular rational function $R\left(z\right)$ such that $F\left(z\right)/R\left(z\right)$ 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 $R\left(z\right)$ can be taken to be a polynomial ${z}^{\gamma }Q\left(z\right)$ for some explicit non-negative integer $\gamma$ and such that $1/Q\left(z\right)$ is $k$-regular. The article was published this year.