Application of discrete geometry to the construction of Laurent-rational zeros

We consider zeros of polynomials whose coefficients lie in the field C((t)) of formal Laurent series with complex coefficients. The algebraic closure of C((t)) is the field K of “Puiseux series,” which allow fractional exponents. There is a well-known algorithm, described in [9], for constructing roots in K to a one-variable polynomial over C((t)). Several papers give generalizations of this algorithm; for instance, [5] gives an algorithm for constructing zeros to systems of multivariable polynomials over C((t)). We generalize the one-variable algorithm to multivariable polynomials with the specific goal of bounding the degree of the field extension over C((t)) in which the specified zeros lie. We adapt recent techniques from tropical geometry which involve discrete and piecewiselinear geometry.