Laws of inertia in higher degree binary forms
Author:
Bruce Reznick
Journal:
Proc. Amer. Math. Soc. 138 (2010), 815826
MSC (2010):
Primary 11E76, 15A21
Published electronically:
November 3, 2009
MathSciNet review:
2566547
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Abstract: We consider representations of real forms of even degree as a linear combination of powers of real linear forms, counting the number of positive and negative coefficients. We show that the natural generalization of Sylvester's Law of Inertia holds for binary quartics but fails for binary sextics.
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B. Reznick, The length of binary forms, in preparation.
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J. J. Sylvester, On a remarkable discovery in the theory of canonical forms and of hyperdeterminants, originally in Philosophical Magazine, vol. 2, 1851; Paper 42 in Mathematical Papers, Vol. 1, Chelsea, New York, 1973. Originally published by Cambridge University Press in 1904.
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J. J. Sylvester, On an elementary proof and generalization of Sir Isaac Newton's hitherto undemonstrated rule for the discovery of imaginary roots, Proc. Lond. Math. Soc. 1 (1865/1866), 116; Paper 84 in Mathematical Papers, Vol. 2, Chelsea, New York, 1973. Originally published by Cambridge University Press in 1908.
 1.
 S. Gundelfinger, Zur Theorie der binären Formen, J. Reine Angew. Math., 100 (1886), 413424.
 2.
 J. P. S. Kung, Gundelfinger's theorem on binary forms, Stud. Appl. Math., 75 (1986), 163169. MR 859177 (87m:11020)
 3.
 J. P. S. Kung and G.C. Rota, The invariant theory of binary forms, Bull. Amer. Math. Soc. (N.S.), 10 (1984), 2785. MR 722856 (85g:05002)
 4.
 G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. II, SpringerVerlag, New York, 1976. MR 0465631 (57:5529)
 5.
 V. Powers and B. Reznick, Notes towards a constructive proof of Hilbert's theorem on ternary quartics, Quadratic forms and their applications, Dublin, 1999 (A. Ranicki, ed.), Contemp. Math., vol. 272, Amer. Math. Soc., Providence, RI, 2000, pp. 209227. MR 1803369 (2001h:11049)
 6.
 B. Reznick, Sums of even powers of real linear forms, Mem. Amer. Math. Soc. 96, no. 463 (1992). MR 1096187 (93h:11043)
 7.
 B. Reznick, Homogeneous polynomial solutions to constant coefficient PDE's, Adv. Math., 117 (1996), 179192. MR 1371648 (97a:12006)
 8.
 B. Reznick, The length of binary forms, in preparation.
 9.
 J.J. Sylvester, An Essay on Canonical Forms, Supplement to a Sketch of a Memoir on Elimination, Transformation and Canonical Forms, originally published by George Bell, Fleet Street, London, 1851; Paper 34 in Mathematical Papers, Vol. 1, Chelsea, New York, 1973. Originally published by Cambridge University Press in 1904.
 10.
 J. J. Sylvester, On a remarkable discovery in the theory of canonical forms and of hyperdeterminants, originally in Philosophical Magazine, vol. 2, 1851; Paper 42 in Mathematical Papers, Vol. 1, Chelsea, New York, 1973. Originally published by Cambridge University Press in 1904.
 11.
 J. J. Sylvester, On an elementary proof and generalization of Sir Isaac Newton's hitherto undemonstrated rule for the discovery of imaginary roots, Proc. Lond. Math. Soc. 1 (1865/1866), 116; Paper 84 in Mathematical Papers, Vol. 2, Chelsea, New York, 1973. Originally published by Cambridge University Press in 1908.
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Additional Information
Bruce Reznick
Affiliation:
Department of Mathematics, University of Illinois at UrbanaChampaign, Urbana, Illinois 61801
Email:
reznick@math.uiuc.edu
DOI:
http://dx.doi.org/10.1090/S0002993909101867
Received by editor(s):
June 30, 2009
Published electronically:
November 3, 2009
Communicated by:
Ken Ono
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
