Positive polynomials and hyperdeterminants
type: Item
instance of: scholarly article
title: Positive polynomials and hyperdeterminants
zbMATH Open document ID: 1134.13020
author: Fernando Cukierman
published in: Collectanea Mathematica
publication date: 2008-03-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0312469
full work available at URL: https://eudml.org/doc/42037
review text: Let \(F\) be a homogeneous real polynomial of even degree in any number of variables. Here the author considers the problem of giving explicit conditions on the coefficients so that \(F\) is positive definite or positive semi-definite. He gives a necessary condition for positivity, and a sufficient condition for non-negativity, in terms of positivity or semipositivity of a one variable characteristic polynomial. He translates the sufficient condition in terms of Hankel matrices. Conditions are known for degree \(2\) (quadratic forms), and for two variables (binary form). He gives a clever reduction of the general case to these cases. Inside the vector space of all real homogeneous polynomials with fixed degree and number of variables there are the cone \(\Pi\) of positive polynomials and the discriminant hypersurface \(\Delta\). Part of his work is the study of their mutual geometry (connectedness of \(\Pi \backslash \Delta\)).
reviewed by: E. Ballico
Mathematics Subject Classification ID: 13J30
Mathematics Subject Classification ID: 14P99
Mathematics Subject Classification ID: 14M12
zbMATH DE Number: 5245337
zbMATH Keywords: positive definite
zbMATH Keywords: discriminant
zbMATH Keywords: hyperderminant
zbMATH Keywords: characteristic polynomial
zbMATH Keywords: positive semi-definite
zbMATH Keywords: Hankel matrices
MaRDI profile type: Publication
arXiv ID: math/0312469
description: scientific article