  We deal with the algebraicity of a Puiseux series in terms of the properties
of its coefficients. We show that the algebraicity of a Puiseux series for
given bounded degree is determined by a finite number of explicit polynomial
formulae. Conversely, given a vanishing polynomial, there is a closed-form
formula for the coefficients of the series in terms of the coefficients of the
polynomial and of an initial part of the series.
