Intersection multiplicity of a sparse curve and a trinomial curve

Source abstract

Let f $\in$ C[x, y] be a trinomial and let g $\in$ C[x, y] be a sparse polynomial with N monomials (N $\ge$ 1). We address the problem of bounding the local intersection multiplicity of the bivariate polynomial system f = g = 0 at an isolated root q $\in$ (C * )^ 2 only in terms of N . Our main result establishes that the local multiplicity of the system at any isolated solution q $\in$ (C * ) ^2 is at most (N -1)(N +2)/ 2 , independently of the degrees of f and g. Furthermore, we know that this bound is sharp for N = 3.