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 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 $X\cup {(X^{\vee }\cap Q)}^{\vee }$.

中文翻译：

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实代数变体的距离函数

对于赋予非退化二次形式q的欧几里得空间V中的任何（实）代数变体X，我们引入一个多项式${\mathrm{EDpoly}}_{X，ü}（{Ť}^2）$ 对于任何 $ü\in V$，其根源在于从u到X的距离。程度${\mathrm{EDpoly}}_{X，ü}$是欧氏距离度的X。当X是一个射影变数时，我们证明了对偶性，即${\mathrm{EDpoly}}_{X，ü}（{Ť}^2）={\mathrm{EDpoly}}_{X^{\vee }，ü}（q（ü）-{Ť}^2）$ 哪里 $X^{\vee }$是X的对偶变体。当X横切成各向同性二次Q时，我们证明X的ED多项式是一元的，其下项的零轨迹为$X\cup {（X^{\vee }\cap 问）}^{\vee }$。