Euclidean distance degree of complete intersections via Newton polytopes

2024/04/26 by Nguyễn Tất Thắng, Thang, Nguyen Tat, Pham Thu Thuy +1 · 2 citations
Engineering · Mathematics · #14M25 #90C26 #Advanced Numerical Analysis Techniques #Advanced Statistical Methods and Models #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2404.17237

openalex publication_date 2024/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note, we consider a complete intersection X=\x∈ ℝⁿ : f₁(x)= … = 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,…, 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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