Norm Minimization, Invariant Theory, and the Jacobian conjecture

Consider the action of a group on a finite-dimensional vector space. Given some natural conditions on the group, Hilbert showed a famous "duality" between invariant polynomials and closures of group orbits. Namely, the orbit closure of a vector is separated from the origin if and only if some homogeneous invariant polynomial has a nonzero value on the vector.

We prove a quantitative sharpening of this duality, relating the magnitudes of invariant polynomials evaluated on the vector to the minimal distance between the orbit closure and the origin; the latter quantity is known as the capacity in the matrix/operator/tensor scaling literature.

Our result has interpretations in quantum tomography, such as: given many identical copies of a quantum particle, what's the chance that a measurement of their overall angular momentum yields zero? Our result and its proof are also related to a famous problem in algebraic geometry known as the Jacobian conjecture.

The talk will assume no special background knowledge - I hope to explain the many notions and connections above. It is based on the joint work https://arxiv.org/abs/2004.14872 with Michael Walter.