The Euclidean Distance Degree of an Algebraic Variety

The nearest point map of a real algebraic variety with respect to euclidean distance is an algebraic function. for instance, for varieties of low-rank matrices, the eckart–young theorem states that this map is given by the singular value decomposition. this article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. the euclidean distance degree of a variety is the number of critical points of the squared distance to a general point outside the variety. focusing on varieties seen in applications, we present numerous tools for exact computations.