Euclidean distance degree of complete intersections via Newton polytopes

TL;DR

This paper establishes a method to compute the Euclidean distance degree of complete intersections in real space using the mixed volume of their Newton polytopes, generalizing previous results for the case of a single polynomial.

Contribution

It extends the computation of Euclidean distance degree to complete intersections via Newton polytopes, broadening the applicability of earlier single-polynomial results.

Findings

Euclidean distance degree can be expressed via mixed volume of Newton polytopes.

The result applies when Newton polytopes contain the origin and polynomials are generic.

Generalizes previous work from single polynomial to complete intersections.

Abstract

In this note, we consider a complete intersection $X=\{x\in \mathbb{R}^n : f_1(x)= \ldots = f_m(x)=0\}, n>m$ and study its Euclidean distance degree in terms of the mixed volume of the Newton polytopes. We show that if the Newton polytopes of $f_j,j=1,\ldots, m$ contain the origin then when these polynomials are generic with respect to their Newton polytopes, the Euclidean distance degree of $X$ can be computed in terms of the mixed volume of Newton polytopes associated to $f_j$. This is a generalization for the result by P. Breiding, F. Sottile and J. Woodcock in case $m=1$.