Speaker
Luca Sodomaco, MPI MiS Leipzig
Abstract
The (Euclidean) distance degree of an algebraic variety X is among the most studied topics in Metric Algebraic Geometry. It equals the number of complex critical points of the squared distance function from a generic point u (a polynomial quadratic objective function) restricted to the locus of nonsingular points of X. The univariate polynomial (in an unknown t) whose roots are the squared distances between u and a critical point x is the (Euclidean) distance polynomial of X. The coefficients of this polynomial are themselves polynomials in the coordinates of u. Evaluating the distance polynomial of X at some value t=ε gives the equation of the ε-offset hypersurface of X.
This talk presents a project with Emil Horobeț on the “”distance polynomial map””, that is, the map sending a point u to the tuple of coefficients of the distance polynomial of X evaluated at u. This talk focuses primarily on the case where X is the cone over the Veronese variety in the space of symmetric tensors, equipped with the Bombieri-Weyl quadratic form. In this setting, the “”distance polynomial”” is a generalization of the “”characteristic polynomial”” of a symmetric matrix, and the image of the “”distance polynomial map”” parametrizes univariate polynomials that are the characteristic polynomial of some symmetric tensor.