|
Laws of inertia in higher degree binary forms
Author(s):
Bruce
Reznick
Journal:
Proc. Amer. Math. Soc.
MSC (2010):
Primary 11E76, 15A21
Posted:
November 3, 2009
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
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.
References:
-
- 1.
- S. Gundelfinger, Zur Theorie der binären Formen, J. Reine Angew. Math., 100 (1886), 413-424.
- 2.
- J. P. S. Kung, Gundelfinger's theorem on binary forms, Stud. Appl. Math., 75 (1986), 163-169. 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), 27-85. MR 722856 (85g:05002)
- 4.
- G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. II, Springer-Verlag, 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. 209-227. 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), 179-192. 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), 1-16; Paper 84 in Mathematical Papers, Vol. 2, Chelsea, New York, 1973. Originally published by Cambridge University Press in 1908.
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2010):
11E76, 15A21
Retrieve articles in all Journals with MSC
(2010):
11E76, 15A21
Additional Information:
Bruce
Reznick
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email:
reznick@math.uiuc.edu
DOI:
10.1090/S0002-9939-09-10186-7
PII:
S 0002-9939(09)10186-7
Received by editor(s):
June 30, 2009
Posted:
November 3, 2009
Communicated by:
Ken Ono
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org