Intersection multiplicity of a sparse curve and a low-degree curve

04/01/2019

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by   Pascal Koiran, et al.

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Let F(x, y) ∈C[x,y] be a polynomial of degree d and let G(x,y) ∈C[x,y] be a polynomial with t monomials. We want to estimate the maximal multiplicity of a solution of the system F(x,y) = G(x,y) = 0. Our main result is that the multiplicity of any isolated solution (a,b) ∈C^2 with nonzero coordinates is no greater than 5/2d^2t^2. We ask whether this intersection multiplicity can be polynomially bounded in the number of monomials of F and G, and we briefly review some connections between sparse polynomials and algebraic complexity theory.