Laws of inertia in higher degree binary forms

Reznick, Bruce

doi:10.1090/S0002-9939-09-10186-7

Laws of inertia in higher degree binary forms
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Proc. Amer. Math. Soc. 138 (2010), 815-826 Request permission

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

S. Gundelfinger, Zur Theorie der binären Formen, J. Reine Angew. Math., 100 (1886), 413–424.
Joseph P. S. Kung, Gundelfinger’s theorem on binary forms, Stud. Appl. Math. 75 (1986), no. 2, 163–169. MR 859177, DOI 10.1002/sapm1986752163
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, DOI 10.1090/S0273-0979-1984-15188-7
G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. II, Revised and enlarged translation by C. E. Billigheimer of the fourth German edition, Springer Study Edition, Springer-Verlag, New York-Heidelberg, 1976. Theory of functions, zeros, polynomials, determinants, number theory, geometry. MR 0465631, DOI 10.1007/978-1-4757-6292-1
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, DOI 10.1090/conm/272/04405
Bruce Reznick, Sums of even powers of real linear forms, Mem. Amer. Math. Soc. 96 (1992), no. 463, viii+155. MR 1096187, DOI 10.1090/memo/0463
Bruce Reznick, Homogeneous polynomial solutions to constant coefficient PDE’s, Adv. Math. 117 (1996), no. 2, 179–192. MR 1371648, DOI 10.1006/aima.1996.0007
B. Reznick, The length of binary forms, in preparation.
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.
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.
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.