Polynomial norms

Amirali Ahmadi, Etienne de Klerk, Georgina Hall

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In this paper, we study polynomial norms, i.e., norms that are the $d$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 $d$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 solving a hierarchy of semidefinite programs. We further study the problem of optimizing over the set of polynomial norms using semidefinite programming. To do this, we introduce the notion of $r$-sum of squares-convexity and extend a result of Reznick on sum of squares representations of positive definite forms to positive definite biforms. We conclude with some applications of polynomial norms to statistics and dynamical systems.
Original languageEnglish
Pages (from-to)399–422
JournalSIAM Journal on Optimization
Issue number1
Publication statusPublished - 2019


  • polynomial norms
  • sum of squares
  • polynomials
  • convex polynomials
  • semidefinitie programming


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