Proceedings of the American Mathematical Society

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

   
 

 

Pfister's theorem fails in the Hermitian case


Authors: John P. D’Angelo and Jiří Lebl
Journal: Proc. Amer. Math. Soc. 140 (2012), 1151-1157
MSC (2010): Primary 12D15, 14P05, 15B57, 32V15
Published electronically: April 1, 2011
MathSciNet review: 2869101
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Abstract: We show that the Hermitian analogue of a famous result of Pfister fails. To do so we provide a Hermitian symmetric polynomial $ r$ of total degree $ 2d$ such that any nonzero multiple of it cannot be written as a Hermitian sum of squares with fewer than $ d+1$ squares.


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  • [A] Artin, E., Über die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Univ. Hamburg 5 (1927), 110-115.
  • [D1] John P. D’Angelo, Inequalities from complex analysis, Carus Mathematical Monographs, vol. 28, Mathematical Association of America, Washington, DC, 2002. MR 1899123
  • [D2] D'Angelo, J., Hermitian analogues of Hilbert's $ 17$-th problem, Advances in Math., 226 (2011), 4607-4637.
  • [D3] John P. D’Angelo, Complex variables analogues of Hilbert’s seventeenth problem, Internat. J. Math. 16 (2005), no. 6, 609–627. MR 2153486, 10.1142/S0129167X05002990
  • [DL] D'Angelo, J. and Lebl, J., Hermitian symmetric polynomials and CR complexity, Journal Geometric Analysis (2010) (to appear).
  • [DLP] John P. D’Angelo, Jiří Lebl, and Han Peters, Degree estimates for polynomials constant on a hyperplane, Michigan Math. J. 55 (2007), no. 3, 693–713. MR 2372622, 10.1307/mmj/1197056463
  • [H] Xiaojun Huang, On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions, J. Differential Geom. 51 (1999), no. 1, 13–33. MR 1703603
  • [L] Lebl, J., Normal forms, Hermitian operators, and CR maps of spheres and hyperquadrics, to appear in Michigan Math J., arXiv:0906.0325.
  • [PF] Albrecht Pfister, Zur Darstellung definiter Funktionen als Summe von Quadraten, Invent. Math. 4 (1967), 229–237 (German). MR 0222043
  • [Q] Daniel G. Quillen, On the representation of hermitian forms as sums of squares, Invent. Math. 5 (1968), 237–242. MR 0233770
  • [S] Claus 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. MR 2500469, 10.1007/978-0-387-09686-5_8

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Additional Information

John P. D’Angelo
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, Illinois 61801
Email: jpda@math.uiuc.edu

Jiří Lebl
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0112, La Jolla, California 92093-0112
Email: jlebl@math.uiuc.edu, jlebl@math.ucsd.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10841-4
Keywords: Hilbert’s $17$-th problem, Hermitian forms, sums of squares, Hermitian length, Huang lemma.
Received by editor(s): July 6, 2010
Received by editor(s) in revised form: October 8, 2010, and December 22, 2010
Published electronically: April 1, 2011
Communicated by: Franc Forstneric
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.