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

Pascal Koiran (1) , Mateusz Skomra (2)

1 LIP - Laboratoire de l'Informatique du Parallélisme
2 TROPICAL - TROPICAL

Let $F(x, y)∈C[x, y]$ be a polynomial of degreed 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 $(a, b)∈C2$ with nonzero coordinates is no greater than $52d2t2$. We ask whether this intersection multiplicity can be polynomially boundedin the number of monomials of F and G, and we briefly review someconnections between sparse polynomials and algebraic complexity the-ory.

Domaines: Complexité [cs.CC]

Pascal Koiran, Mateusz Skomra. Intersection multiplicity of a sparse curve and a low-degree curve. Journal of Pure and Applied Algebra, 2020, 224 (7), pp.106279. ⟨10.1016/j.jpaa.2019.106279⟩. ⟨hal-02502580⟩