Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Polynomials non-negative on a strip


Author: M. Marshall
Journal: Proc. Amer. Math. Soc. 138 (2010), 1559-1567
MSC (2010): Primary 14P99; Secondary 12D15, 12E05
DOI: https://doi.org/10.1090/S0002-9939-09-10016-3
Published electronically: December 22, 2009
MathSciNet review: 2587439
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that if $ f(x,y)$ is a polynomial with real coefficients which is non-negative on the strip $ [0,1]\times \mathbb{R}$, then $ f(x,y)$ has a presentation of the form

$\displaystyle f(x,y) = \sum_{i=1}^k g_i(x,y)^2+\sum_{j=1}^{\ell}h_j(x,y)^2x(1-x),$

where the $ g_i(x,y)$ and $ h_j(x,y)$ are polynomials with real coefficients.


References [Enhancements On Off] (What's this?)

  • 1. J. Cimprič, S. Kuhlmann, M. Marshall, Positivity in power series rings. Adv. Geom., to appear.
  • 2. D. Hilbert, Über die Darstellung definiter Formen als Summe von Formenquadraten. Math. Ann. 32, 342-350 (1888). MR 1510517
  • 3. S. Kuhlmann, M. Marshall, Positivity, sums of squares and the multidimensional moment problem. Trans. Amer. Math. Soc. 354, 4285-4301 (2002). MR 1926876 (2003j:14078)
  • 4. M. Marshall, Positive polynomials and sums of squares, AMS Surveys and Monographs, 146, Amer. Math. Soc. (2008). MR 2383959 (2009a:13044)
  • 5. M. Marshall, Cylinders with compact cross-section and the strip conjecture. Séminaire de Structures Algébriques Ordonnées, Univ. Paris 6 et 7, June 2008.
  • 6. T. Motzkin, The arithmetic-geometric inequalities. In: Inequalities, Proc. Symp. Wright-Patterson AFB, 1965 (O. Shisha, ed.), Academic Press, 205-224 (1967). MR 0223521 (36:6569)
  • 7. D. Plaumann, Bounded polynomials, sums of squares, and the moment problem. PhD Thesis, Konstanz, 2008.
  • 8. V. Powers, Positive polynomials and the moment problem for cylinders with compact cross-section. J. Pure and Applied Alg. 188, 217-226 (2004). MR 2030815 (2004k:14107)
  • 9. V. Powers, B. Reznick, Polynomials positive on unbounded rectangles. In: Positive Polynomials in Control, Lecture Notes in Control and Information Sciences, 312 (D. Henrion, A. Garulli, eds.), Springer-Verlag, 151-163 (2005). MR 2123522 (2005m:14108)
  • 10. V. Powers, C. Scheiderer, The moment problem for non-compact semialgebraic sets. Adv. Geom. 1, 71-88 (2001). MR 1823953 (2002c:14086)
  • 11. G. Sansone, J. Gerretsen, Lectures on the theory of functions of a complex variable, Vol. 2, Wolters-Noordhoff Publishing, Gröningen (1969). MR 0259072 (41:3714)
  • 12. C. Scheiderer, Sums of squares of regular functions on real algebraic varieties. Trans. Amer. Math. Soc. 352, 1039-1069 (2000). MR 1675230 (2000j:14090)
  • 13. C. Scheiderer, Sums of squares on real algebraic curves. Math. Z. 245, 725-760 (2003). MR 2020709 (2004k:14103)
  • 14. C. Scheiderer, Non-existence of degree bounds for weighted sums of squares representations J. of Complexity 21, 823-844 (2005). MR 2182447 (2006k:14117)
  • 15. C. Scheiderer, Sums of squares on real algebraic surfaces. Manuscripta Math. 119, 395-410 (2006). MR 2223624 (2006m:14079)
  • 16. K. Schmüdgen, The moment problem for closed semi-algebraic sets J. Reine Angew. Math. 558, 225-234 (2003). MR 1979186 (2004e:47019)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14P99, 12D15, 12E05

Retrieve articles in all journals with MSC (2010): 14P99, 12D15, 12E05


Additional Information

M. Marshall
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK, Canada, S7N 5E6
Email: marshall@math.usask.ca

DOI: https://doi.org/10.1090/S0002-9939-09-10016-3
Keywords: Positive polynomials, sums of squares, moment problem.
Received by editor(s): June 9, 2008
Received by editor(s) in revised form: April 26, 2009
Published electronically: December 22, 2009
Additional Notes: This research was funded in part by an NSERC Discovery Grant.
Communicated by: Ted Chinburg
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society