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arXiv · 2404.17237

Euclidean distance degree of complete intersections via Newton polytopes

Nguyen Tat Thang·Pham Thu Thuy
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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 $m=1$.

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Nguyen Tat Thang, Pham Thu Thuy. 2024-05-02. Euclidean distance degree of complete intersections via Newton polytopes. https://arxiv.org/abs/2404.17237

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