Abstract
Let p be a nonconstant form in \({\mathbb {R}}[x_1,\ldots ,x_n]\) with \(p(1,\ldots ,1)>0\). If \(p^m\) has strictly positive coefficients for some integer \(m\ge 1\), we show that \(p^m\) has strictly positive coefficients for all sufficiently large m. More generally, for any such p and any form q that is strictly positive on \(({\mathbb {R}}_+)^n{\setminus }\{0\}\), we show that the form \(p^mq\) has strictly positive coefficients for all sufficiently large m. This result can be considered as a strict Positivstellensatz for forms relative to \(({\mathbb {R}}_+)^n{\setminus }\{0\}\). We give two proofs, one based on results of Handelman, the other on techniques from real algebra.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Berr and Th. Wörmann, Positive polynomials on compact sets, Manuscripta Math. 104 (2001), 135–143.
J. Bochnak, M. Coste, and M.-F. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 36, Springer, Berlin, 1998.
S. Burgdorf, C. Scheiderer, and M. Schweighofer, Pure states, nonnegative polynomials and sums of squares, Comment. Math. Helv. 87 (2012), 113–140.
D. Catlin, The Bergman kernel and a theorem of Tian, In: Analysis and Geometry in Several Complex Variables (Katata, 1997), 1–23, Trends Math., Birkhäuser Boston, Boston, MA, 1999.
D. Catlin and J. D’Angelo, An isometric imbedding theorem for holomorphic bundles, Math. Res. Lett. 6 (1999), 43–60.
V. De Angelis, Asymptotic expansions and positivity of coefficients for large powers of analytic functions, Int. J. Math. Sci. 16 (2003), 1003–1025.
J. D’Angelo and D. Varolin, Postivity conditions for Hermitian symmetric functions, Asian J. Math. 8 (2004), 215-232.
D. Handelman, Deciding eventual positivity of polynomials, Ergodic Theory Dynam. Systems 6 (1986), 342–350.
D. Handelman, Polynomials with a positive power, In: Symbolic dynamics and its applications, P. Walters (ed.), Contemp. Math. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 229–230.
D. Handelman, Stability of real polynomials with positive coefficients, Answer on MathOverflow, URL (version: 2014-09-10): http://mathoverflow.net/q/180475.
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, 2nd ed., Cambridge University Press, Cambridge, 1952.
J.-L. Krivine, Quelques propriétés des préordres dans les anneaux commutatifs unitaires, C. R. Acad. Sc. Paris 258 (1964), 3417–3418.
M. Marshall, Positive Polynomials and Sums of Squares, Math. Surveys and Monographs 146, Amer. Math. Soc., Providence, RI, 2008.
G. Pólya, Über positive Darstellung von Polynomen, Vierteljschr. Naturforsch. Ges. Zürich 73 (1928), 141–145, In: Collected Papers 2, MIT Press, 1974, 309–313.
C. Scheiderer, Sums of squares on real algebraic surfaces, Manuscripta Math. 119 (2006), 395–410.
C. Scheiderer, A Positivstellensatz for projective real varieties, Manuscripta Math. 138 (2012), 73–88.
M. Schweighofer, Certificates for nonnegativity of polynomials with zeros on compact semialgebraic sets, Manuscripta Math. 117 (2005), 407–428.
C. Tan and W.-K. To, Characterization of polynomials whose large powers have all positive coefficients, Preprint, arXiv:1701.02040, to appear in Proc. Amer. Math. Soc.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Scheiderer, C., Tan, C. A Positivstellensatz for forms on the positive orthant. Arch. Math. 109, 123–131 (2017). https://doi.org/10.1007/s00013-017-1054-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-017-1054-z