The Real Vanishing Ideals of Nuclear p-Norm Balls

Abstract

We study the algebraic and geometric structure related to tensor nuclear norms. We show that the unit ball of the nuclear norm is the convex hull of an irreducible real variety and give an explicit description of its real vanishing ideal. As a consequence, we obtain a simple criterion to decide when a primary ideal is prime, and we use it to prove that the ideal of the nuclear 2-norm is real reduced and prime.