The Distance Function from a Real Algebraic Variety

Abstract: For any (real) algebraic variety $X$ in a Euclidean space $V$ endowed with a nondegenerate quadratic form $q$, we introduce a polynomial $\mathrm{EDpoly}<em>{X,u}(t<sup>2)$ which, for any $u\in V$, has among its roots the distance from $u$ to $X$. The degree of $\mathrm{EDpoly}</em>{X,u}$ is the {\em Euclidean Distance degree} of $X$. We prove a duality property when $X$ is a projective variety, namely $$\mathrm{EDpoly}<em>{X,u}(t^{2)=\mathrm{EDpoly}}</em>{X<sup>\vee,u}(q(u)-t<sup>2)$$ where $X<sup>\vee$ 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 $X\cup(X^\vee\cap Q)<sup>\vee$.