Polynomial Norms

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

Bounded decomposition in the Brieskorn lattice and Pfaffian Picard--Fuchs systems for Abelian integrals

We suggest an algorithm for derivation of the Picard--Fuchs system of Pfaffian equations for Abelian integrals corresponding to semiquasihomogeneous Hamiltonians. It is based on an effective decomposition of polynomial forms in the Brieskorn lattice. The construction allows for an explicit upper bound on the norms of the polynomial coefficients, an important ingredient in studying zeros of these integrals.Comment: 17 pages in LaTeX2

Unconditionality in tensor products and ideals of polynomials, multilinear forms and operators

We study tensor norms that destroy unconditionality in the following sense: for every Banach space $E$ with unconditional basis, the $n$-fold tensor product of $E$ (with the corresponding tensor norm) does not have unconditional basis. We establish an easy criterion to check weather a tensor norm destroys unconditionality or not. Using this test we get that all injective and projective tensor norms different from $\varepsilon$ and $\pi$ destroy unconditionality, both in full and symmetric tensor products. We present applications to polynomial ideals: we show that many usual polynomial ideals never enjoy the Gordon-Lewis property. We also consider the unconditionality of the monomial basic sequence. Analogous problems for multilinear and operator ideals are addressed.Comment: 23 page

Polynomial Threshold Functions, AC^0 Functions and Spectral Norms

The class of polynomial-threshold functions is studied using harmonic analysis, and the results are used to derive lower bounds related to AC^0 functions. A Boolean function is polynomial threshold if it can be represented as a sign function of a sparse polynomial (one that consists of a polynomial number of terms). The main result is that polynomial-threshold functions can be characterized by means of their spectral representation. In particular, it is proved that a Boolean function whose L_1 spectral norm is bounded by a polynomial in n is a polynomial-threshold function, and that a Boolean function whose L_∞^(-1) spectral norm is not bounded by a polynomial in n is not a polynomial-threshold function. Some results for AC^0 functions are derived