Euclidean distance degree of complete intersections via Newton polytopes
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911862519496704 |
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| author | Thang, Nguyen Tat Thuy, Pham Thu |
| author_facet | Thang, Nguyen Tat Thuy, Pham Thu |
| contents | 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 $m=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17237 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Euclidean distance degree of complete intersections via Newton polytopes Thang, Nguyen Tat Thuy, Pham Thu Algebraic Geometry 14M25, 90C26 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 $m=1$. |
| title | Euclidean distance degree of complete intersections via Newton polytopes |
| topic | Algebraic Geometry 14M25, 90C26 |
| url | https://arxiv.org/abs/2404.17237 |