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

## TL;DR

This paper establishes an upper bound on the intersection multiplicity of a sparse polynomial and a low-degree polynomial at solutions with nonzero coordinates, linking algebraic geometry with complexity theory.

## Contribution

It provides a new bound on intersection multiplicity for sparse and low-degree polynomial systems, addressing a key question in algebraic geometry and complexity theory.

## Key findings

- Maximum multiplicity bounded by (5/2)d^2 t^2 for solutions with nonzero coordinates
- Highlights the connection between sparse polynomials and algebraic complexity
- Raises questions about polynomial bounds in monomial counts

## Abstract

Let $F(x, y) \in \mathbb{C}[x,y]$ be a polynomial of degree $d$ and let $G(x,y) \in \mathbb{C}[x,y]$ 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 $(a,b) \in \mathbb{C}^2$ with nonzero coordinates is no greater than $\frac{5}{2}d^2t^2$. 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.