The Distance Function from a Real Algebraic Variety
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866912738430681088 |
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| author | Ottaviani, Giorgio Sodomaco, Luca |
| author_facet | Ottaviani, Giorgio Sodomaco, Luca |
| contents | For any (real) algebraic variety $X$ in a Euclidean space $V$ endowed with a nondegenerate quadratic form $q$, we introduce a polynomial $\mathrm{EDpoly}_{X,u}(t^2)$ which, for any $u\in V$, has among its roots the distance from $u$ to $X$. The degree of $\mathrm{EDpoly}_{X,u}$ is the {\em Euclidean Distance degree} of $X$. We prove a duality property when $X$ is a projective variety, namely $\mathrm{EDpoly}_{X,u}(t^2)=\mathrm{EDpoly}_{X^\vee,u}(q(u)-t^2)$ where $X^\vee$ is the dual variety of $X$. When $X$ is transversal to the isotropic quadric $Q$, we prove that the ED polynomial of $X$ is monic and the zero locus of its lower term is $X\cup(X^\vee\cap Q)^\vee$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1807_10390 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | The Distance Function from a Real Algebraic Variety Ottaviani, Giorgio Sodomaco, Luca Algebraic Geometry 14C17, 14N07, 14P05, 15A72, 58K05, 90C26 For any (real) algebraic variety $X$ in a Euclidean space $V$ endowed with a nondegenerate quadratic form $q$, we introduce a polynomial $\mathrm{EDpoly}_{X,u}(t^2)$ which, for any $u\in V$, has among its roots the distance from $u$ to $X$. The degree of $\mathrm{EDpoly}_{X,u}$ is the {\em Euclidean Distance degree} of $X$. We prove a duality property when $X$ is a projective variety, namely $\mathrm{EDpoly}_{X,u}(t^2)=\mathrm{EDpoly}_{X^\vee,u}(q(u)-t^2)$ where $X^\vee$ is the dual variety of $X$. When $X$ is transversal to the isotropic quadric $Q$, we prove that the ED polynomial of $X$ is monic and the zero locus of its lower term is $X\cup(X^\vee\cap Q)^\vee$. |
| title | The Distance Function from a Real Algebraic Variety |
| topic | Algebraic Geometry 14C17, 14N07, 14P05, 15A72, 58K05, 90C26 |
| url | https://arxiv.org/abs/1807.10390 |