The Real Vanishing Ideals of Nuclear p-Norm Balls

TL;DR

This paper explores the algebraic and geometric properties of tensor nuclear norm balls, providing explicit descriptions of their vanishing ideals and criteria for primality, advancing understanding in tensor norm geometry.

Contribution

It offers an explicit description of the real vanishing ideal of tensor nuclear norm balls and establishes primality conditions for related ideals, which is a novel theoretical insight.

Findings

The unit ball of the nuclear norm is the convex hull of an irreducible real variety.

The paper provides an explicit description of the real vanishing ideal of nuclear norm balls.

The ideal of the nuclear 2-norm is shown to be real reduced and prime.

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.