  We deal with the algebraicity of an iterated Puiseux series in several
variables in terms of the properties of its coefficients. Our aim is to
generalize to several variables the results from [HM15]. We show that the
algebraicity of such a series for given bounded degrees is determined by a
finite number of explicit universal polynomial formulas. 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 a bounded
initial part of the series.
