  It is classical that univariate algebraic functions satisfy linear
differential equations with polynomial coefficients. Linear recurrences follow
for the coefficients of their power series expansions. We show that the linear
differential equation of minimal order has coefficients whose degree is cubic
in the degree of the function. We also show that there exists a linear
differential equation of order linear in the degree whose coefficients are only
of quadratic degree. Furthermore, we prove the existence of recurrences of
order and degree close to optimal. We study the complexity of computing these
differential equations and recurrences. We deduce a fast algorithm for the
expansion of algebraic series.
