Illinois Journal of Mathematics

Hermitian algebra on the ellipse

Mihai Putinar and Claus Scheiderer

Full-text: Open access

Abstract

The subtle distinction between hermitian sums of squares and sums of squares, regarded as positivity certificates of a polynomial restricted to a real algebraic variety, is analyzed on the simplest, yet very relevant, example: an ellipse.

Article information

Source
Illinois J. Math., Volume 56, Number 1 (2012), 213-220.

Dates
First available in Project Euclid: 27 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1380287468

Digital Object Identifier
doi:10.1215/ijm/1380287468

Mathematical Reviews number (MathSciNet)
MR3117026

Zentralblatt MATH identifier
1296.12001

Subjects
Primary: 12D15: Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) [See also 11Exx]
Secondary: 14P05: Real algebraic sets [See also 12D15, 13J30] 15B57: Hermitian, skew-Hermitian, and related matrices 32V15: CR manifolds as boundaries of domains

Citation

Putinar, Mihai; Scheiderer, Claus. Hermitian algebra on the ellipse. Illinois J. Math. 56 (2012), no. 1, 213--220. doi:10.1215/ijm/1380287468. https://projecteuclid.org/euclid.ijm/1380287468


Export citation

References

  • A. Athavale, Holomorphic kernels and commuting operators, Trans. Amer. Math. Soc. 304 (1987), 101–110.
  • D. W. Catlin and J. P. D'Angelo, A stabilization theorem for Hermitian forms and applications to holomorphic mappings, Math. Res. Lett. 3 (1996), 149–166.
  • J. P. D'Angelo and M. Putinar, Hermitian complexity of real polynomial ideals, Internat. J. Math. 23 (2012), 1250026 (14 pages).
  • M. G. Krein, Über eine neue Klasse von hermitischen Formen und über eine Verallgemeinerung des trigonometrischen Momentproblems, Izvestia Akad. Nauk. 9 (1933), 1259–1275.
  • A. Prestel and Ch. N. Delzell, Positive polynomials, Springer, Berlin, 2001.
  • M. Putinar and C. Scheiderer, Sums of Hermitian squares on pseudoconvex boundaries, Math. Res. Lett. 17 (2010), 1047–1053.
  • M. Putinar and C. Scheiderer, Quillen property of real algebraic varieties, preprint; available at http://arxiv.org/abs/1304.0947.
  • D. G. Quillen, On the representation of Hermitian forms as sums of squares, Invent. Math. 5 (1968), 237–242.
  • C. Scheiderer, Positivity and sums of squares: A guide to recent results, Emerging applications of algebraic geometry, IMA Vol. Math. Appl., vol. 149, Springer, New York, 2009, pp. 271–324.
  • J. Stochel and F. H. Szafraniec, Algebraic operators and moments on algebraic sets, Port. Math. 51 (1994), 25–45.