CRM – SNS.<\/p>\n
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 <\/span>F<\/span>p<\/span>((<\/span>x<\/span>\u2212<\/span>1<\/span>)), where <\/span>p <\/span>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\u2019s theorem is best possible in <\/span>F<\/span>p<\/span>((<\/span>x<\/span>\u2212<\/span>1<\/span>)). Recently, in a joint work with Han (built on a previous work by Han and Hu), we proved that, for any distinct nonconstant polynomials <\/span>a,b <\/span>in <\/span>F<\/span>2<\/span>[<\/span>x<\/span>], the power series<\/span><\/p>\n [<\/span>a<\/span>; <\/span>b, b, a, b, a, a, b, . . .<\/span>] = <\/span>a <\/span>+ 1\/(b+1\/b…)<\/span><\/p>\n whose sequence of partial quotients is given by the Thue\u2013Morse sequence, is algebraic of degree 4 over <\/span>F<\/span>2<\/span>(<\/span>x<\/span>). We discuss this and related results. Furthermore, we give a complete description of the continued fraction expansion of the algebraic power series (1 + <\/span>x<\/span>\u2212<\/span>1<\/span>)<\/span>j\/d <\/span>in <\/span>F<\/span>p<\/span>((<\/span>x<\/span>\u2212<\/span>1<\/span>)), where <\/span>j,d <\/span>are coprime integers with <\/span>d <\/span>\u2265 <\/span>3, 1 <\/span>\u2264 <\/span>j < d\/<\/span>2, and gcd(<\/span>p, jd<\/span>) = 1 (joint work with Han).<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n