# Polynomial Norms

## TL;DR

This paper explores polynomial norms, characterizing their properties, computational complexity, and approximation capabilities, and introduces new methods for optimization and applications in statistics and dynamical systems.

## Contribution

It provides a complete characterization of polynomial norms, analyzes their computational hardness, and develops semidefinite programming techniques for their optimization.

## Key findings

- Polynomial norms are characterized by strict convexity of the underlying polynomial.
- Testing whether a form defines a polynomial norm is strongly NP-hard for degree 4.
- Polynomial norms can be approximated arbitrarily well by general norms.

## Abstract

In this paper, we study polynomial norms, i.e. norms that are the $d^{\text{th}}$ root of a degree-$d$ homogeneous polynomial $f$. We first show that a necessary and sufficient condition for $f^{1/d}$ to be a norm is for $f$ to be strictly convex, or equivalently, convex and positive definite. Though not all norms come from $d^{\text{th}}$ roots of polynomials, we prove that any norm can be approximated arbitrarily well by a polynomial norm. We then investigate the computational problem of testing whether a form gives a polynomial norm. We show that this problem is strongly NP-hard already when the degree of the form is 4, but can always be answered by testing feasibility of a semidefinite program (of possibly large size). We further study the problem of optimizing over the set of polynomial norms using semidefinite programming. To do this, we introduce the notion of r-sos-convexity and extend a result of Reznick on sum of squares representation of positive definite forms to positive definite biforms. We conclude with some applications of polynomial norms to statistics and dynamical systems.