Laws of inertia in higher degree binary forms

Author:
Bruce Reznick

Journal:
Proc. Amer. Math. Soc. **138** (2010), 815-826

MSC (2010):
Primary 11E76, 15A21

DOI:
https://doi.org/10.1090/S0002-9939-09-10186-7

Published electronically:
November 3, 2009

MathSciNet review:
2566547

Full-text 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.

**1.**S. Gundelfinger,*Zur Theorie der binären Formen*, J. Reine Angew. Math.,**100**(1886), 413-424.**2.**Joseph P. S. Kung,*Gundelfinger’s theorem on binary forms*, Stud. Appl. Math.**75**(1986), no. 2, 163–169. MR**859177**, https://doi.org/10.1002/sapm1986752163**3.**Joseph P. S. Kung and Gian-Carlo Rota,*The invariant theory of binary forms*, Bull. Amer. Math. Soc. (N.S.)**10**(1984), no. 1, 27–85. MR**722856**, https://doi.org/10.1090/S0273-0979-1984-15188-7**4.**G. Pólya and G. Szegő,*Problems and theorems in analysis. Vol. II*, Springer Study Edition, Springer-Verlag, New York-Heidelberg, 1976. Theory of functions, zeros, polynomials, determinants, number theory, geometry. MR**0465631****5.**Victoria Powers and Bruce Reznick,*Notes towards a constructive proof of Hilbert’s theorem on ternary quartics*, Quadratic forms and their applications (Dublin, 1999) Contemp. Math., vol. 272, Amer. Math. Soc., Providence, RI, 2000, pp. 209–227. MR**1803369**, https://doi.org/10.1090/conm/272/04405**6.**Bruce Reznick,*Sums of even powers of real linear forms*, Mem. Amer. Math. Soc.**96**(1992), no. 463, viii+155. MR**1096187**, https://doi.org/10.1090/memo/0463**7.**Bruce Reznick,*Homogeneous polynomial solutions to constant coefficient PDE’s*, Adv. Math.**117**(1996), no. 2, 179–192. MR**1371648**, https://doi.org/10.1006/aima.1996.0007**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.

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Additional Information

**Bruce Reznick**

Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

Email:
reznick@math.uiuc.edu

DOI:
https://doi.org/10.1090/S0002-9939-09-10186-7

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.