# A geometric criterion on the equality between BKK bound and intersection index

## TL;DR

This paper establishes a simple geometric criterion to determine when the BKK bound equals the intersection index for systems of Laurent polynomials, with applications to algebraic Kuramoto equations.

## Contribution

It introduces a geometric condition that characterizes the equality between the BKK bound and the intersection index for Laurent polynomial systems.

## Key findings

- The intersection index for algebraic Kuramoto equations equals their BKK bound.
- A geometric criterion for the equality between BKK bound and intersection index.
- Application of the criterion to specific polynomial systems.

## Abstract

The Bernshtein-Kushnirenko-Khovanskii theorem provides a generic root count for system of Laurent polynomials in terms of the mixed volume of their Newton polytopes (i.e., the BKK bound). A recent and far-reaching generalization of this theorem is the study of birationally invariant intersection index by Kaveh and Khovanskii. This short note establishes a simple geometric condition on the equality between the BKK bound and the intersection index for a system of vector spaces of Laurent polynomials. Applying this, we show that the intersection index for the algebraic Kuramoto equations equals their BKK bound.