**abstract:**
Almost nothing is known on the continued fraction expansion of an algebraic real number of
degree at least three. The situation is different over the field of power series ${{\mathbb
F}}_{p}((x^{{}-1}))$, where $p$ is a prime number. For instance, there are algebraic power
series of degree at least three whose sequence of partial quotients have bounded degree.
And there are as well algebraic power series of degree at least three which are very well
approximable by rational fractions: the analogue of Liouville's theorem is best possible in
${{\mathbb F}}_{p}((x^{{}-1}))$. Recently, in a joint work with Han (built on a previous work by
Han and Hu), we proved that, for anydistinct nonconstant polynomials $a, b$ in ${{\mathbb
F}}_{2} x$, the power series $$a; b, b, a, b, a, a, b, \ldots = a + \frac{1}{b + \frac{1}{b +
\cdots}} ,$$ whose sequence of partial quotients is given by the Thue--Morse sequence, is
algebraic of degree $4$ over ${{\mathbb F}}_{2} (x)$.We discuss this and related results.
Furthermore, we give a complete description of the continued fraction expansion of the
algebraic power series $(1 + x^{{}-1})^{{jd}$}* in ${{\mathbb F}} _{p}((x^{{}-1}))$, where $j, d$ are
coprime integers with $d \ge 3$, $1 \le j < d*2$, and $\gcd(p, jd) = 1$ (joint work with Han).

Thu 11 May, 15:00 - 16:00, Sala Conferenze Centro De Giorgi

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