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2013 | OriginalPaper | Chapter

On the Length of Binary Forms

Author : Bruce Reznick

Published in: Quadratic and Higher Degree Forms

Publisher: Springer New York

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Abstract

The K-length of a form f in \(K[x_{1},\ldots,x_{n}]\), \(K \subset \mathbb{C}\), is the smallest number of d-th powers of linear forms of which f is a K-linear combination. We present many results, old and new, about K-length, mainly for n = 2, and often about the length of the same form over different fields. For example, the K-length of \(3{x}^{5} - 20{x}^{3}{y}^{2} + 10x{y}^{4}\) is three for \(K = \mathbb{Q}(\sqrt{-1})\), four for \(K = \mathbb{Q}(\sqrt{-2})\) and five for \(K = \mathbb{R}\).

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previous chapter Some Aspects of the Algebraic Theory of Quadratic Forms

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J. Alexander and A. Hirschowitz, Polynomial interpolation in several variables, J. Algebraic Geom., 4 (1995), 201–222, MR1311347 (96f:14065).

A. Białynicki-Birula and A. Schinzel, Extreme binary forms, Acta Arith. 142 (2010), 219–249, MR2606966 (2011d:12001).

G. Blekherman, personal correspondence.

M. Boij, E. Carlini, A. Geramita, Monomials as sums of powers: the real binary case, Proc. Amer. Math. Soc. 139 (2011), 3039–3043, arXiv:1005.3050, MR2811260 (2012e:11070).

M. Brambilla and G. Ottaviani, On the Alexander-Hirschowitz theorem, J. Pure Appl. Algebra, 212 (2008), 1229–1251, arXiv:math/0701409, MR2387598 (2008m:14104).

E. Brown, \({x}^{4} + d{x}^{2}{y}^{2} + {y}^{4} = {z}^{2}\) : some cases with only trivial solutions—and a solution Euler missed, Glasgow Math. J., 31 (1989), 297–307, MR1021805 (91d:11026).

E. Carlini, Varieties of simultaneous sums of power for binary forms, Matematiche (Catania), 57 (2002), 83–97, arXiv:math.AG/0202050, MR2075735 (2005d:11058).

E. Carlini and J. Chipalkatti, On Waring’s problem for several algebraic forms, Comment. Math. Helv., 78 (2003), 494–517, arXiv:math.AG/0112110, MR1998391 (2005b:14097).

A. Causa and R. Re, On the maximum rank of a real binary form, Annali di Matematica Pura et Applicata (4) 190 (2011), no. 1, 55–59, arXiv:1006.5127, MR2747463 (2011k:12001).

10.

M. D. Choi, Z. D. Dai, T. Y. Lam and B. Reznick, The Pythagoras number of some affine algebras and local algebras, J. Reine Angew. Math., 336 (1982), 45–82, MR0671321 (84f:12012).

11.

M. D. Choi, T. Y. Lam, A. Prestel and B. Reznick, Sums of 2mth powers of rational functions in one variable over real closed fields, Math. Z., 221 (1996), 93–112, MR1369464 (96k:12003).

12.

M. D. Choi, T. Y. Lam and B. Reznick, Sums of squares of real polynomials, K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), 103–126, Proc. Sympos. Pure Math., 58, Part 2, Amer. Math. Soc., Providence, RI, 1995, MR1327293 (96f:11058).

13.

J. H. E. Cohn, On the Diophantine equation \({z}^{2} = {x}^{4} + D{x}^{2}{y}^{2} + {y}^{4}\), Glasgow Math. J., 36 (1994), 283–285, MR1295501 (95k:11035).

14.

P. Comon and B. Mourrain, Decomposition of quantics in sums of powers of linear forms, Signal Processing, 53 (1996), 93–107.MATHCrossRef

15.

P. Comon and G. Ottaviani, On the typical rank of real binary forms, Linear Multilinear Algebra, 60 (2012), 657–667, MR 2929176, (arXiv:0909.4865).

16.

L. E. Dickson, History of the Theory of Numbers, vol II: Diophantine Analysis, Carnegie Institute, Washington 1920, reprinted by Chelsea, New York, 1971, MR0245500 (39 #6807b).

17.

I. Dolgachev and V. Kanev, Polar covariants of plane cubics and quartics, Adv. Math., 98 (1993), 216–301, MR1213725 (94g:14029).

18.

R. Ehrenborg and G.-C. Rota, Apolarity and canonical forms for homogeneous polynomials, European J. Combin., 14 (1993), 157–181, MR1215329 (94e:15062).

19.

W. J. Ellison, A ‘Waring’s problem’ for homogeneous forms, Proc. Cambridge Philos. Soc., 65 (1969), 663–672, MR0237450 (38 #5732).

20.

W. J. Ellison, Waring’s problem, Amer. Math. Monthly, 78 (1971), 10–36, MR0414510 (54 #2611).

21.

A. Friedman, Mean-values and polyharmonic polynomials, Michigan Math. J., 4 (1957), 67–74, MR0084045 (18,799b).

22.

A. Geramita, Inverse systems of fat points: Waring’s problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein ideals, The Curves Seminar at Queen’s, Vol. X (Kingston, ON, 1995), 2–114, Queen’s Papers in Pure and Appl. Math., 102, Queen’s Univ., Kingston, ON, 1996, MR1381732 (97h:13012).

23.

S. Gundelfinger, Zur Theorie der binären Formen, J. Reine Angew. Math., 100 (1886), 413–424.

24.

J. Harris, Algebraic geometry. A first course, Graduate Texts in Mathematics, 133, Springer-Verlag, New York, 1992, MR1182558 (93j:14001).

25.

U. Helmke, Waring’s problem for binary forms, J. Pure Appl. Algebra, 80 (1992), 29–45, MR1167385 (93e:11057).

26.

D. Hilbert, Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem), Math. Ann., 67 (1909), 281–300, Ges. Abh. 1, 510–527, Springer, Berlin, 1932, reprinted by Chelsea, New York, 1981.

27.

P. Holgate, Studies in the history of probability and statistics. XLI. Waring and Sylvester on random algebraic equations, Biometrika, 73 (1986), 228–231, MR0836453 (87m:01026).

28.

A. Iarrobino and V. Kanev, Power Sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, 1721 (1999), MR1735271 (2001d:14056).

29.

S. Karlin, Total Positivity, vol. 1, Stanford University Press, Stanford, 1968, MR0230102 (37 #5667).

30.

J. P. S. Kung, Gundelfinger’s theorem on binary forms, Stud. Appl. Math., 75 (1986), 163–169, MR0859177 (87m:11020).

31.

J. P. S. Kung, Canonical forms for binary forms of even degree, in Invariant theory, Lecture Notes in Mathematics, 1278, 52–61, Springer, Berlin, 1987, MR0924165 (89h:15037).

32.

J. P. S. Kung, Canonical forms of binary forms: variations on a theme of Sylvester, in Invariant theory and tableaux (Minnesota, MN, 1988), 46–58, IMA Vol. Math. Appl., 19, Springer, New York, 1990, MR1035488 (91b:11046).

33.

J. P. S. Kung and G.-C. Rota, The invariant theory of binary forms, Bull. Amer. Math. Soc. (N. S.), 10 (1984), 27–85, MR0722856 (85g:05002).

34.

J. M. Landsberg and Z. Teitler, On the ranks and border ranks of symmetric tensors, Found. Comp. Math., 10 (2010), 339–366, arXiv:0901.0487, MR2628829 (2011d:14095).

35.

G. Mehler, Bemerkungen zur Theorie der mechanischen Quadraturen, J. Reine Angew. Math., 63, (1864), 152–157.MATHCrossRef

36.

R. Miranda, Linear systems of plane curves, Notices Amer. Math. Soc., 46 (1999), 192–202, MR1673756 (99m:14012).

37.

L. J. Mordell, Binary cubic forms expressed as a sum of seven cubes of linear forms, J. London. Math. Soc., 42 (1967), 646–651, MR0249355 (40 #2600).

38.

L. J. Mordell, Diophantine equations, Academic Press, London-New York 1969, MR0249355 (40 #2600).

39.

D. J. Newman and M. Slater, Waring’s problem for the ring of polynomials, J. Number Theory, 11 (1979), 477–487, MR0544895 (80m:10016).

40.

I. Niven, Irrational numbers, Carus Mathematical Monographs, No. 11. Math. Assoc. Amer., New York, 1956, MR0080123 (18,195c).

41.

G. Pólya and I. J. Schoenberg, Remarks on de la Vallée Poussin means and convex conformal maps of the circle, Pacific J. Math., 8 (1958), 295–234, MR0100753 (20 #7181).

42.

G. Pólya and G. Szegö, Problems and theorems in analysis, II, Springer-Verlag, New York-Heidelberg 1976, MR0465631 (57 #5529).

43.

V. Powers and B. Reznick, Notes towards a constructive proof of Hilbert’s theorem on ternary quartics, Quadratic forms and their applications (Dublin, 1999), 209–227, Contemp. Math., 272, Amer. Math. Soc., Providence, RI, 2000, MR1803369 (2001h:11049).

44.

K. Ranestad and F.-O. Schreyer, Varieties of sums of powers, J. Reine Angew. Math., 525 (2000), 147–181, MR1780430 (2001m:14009).

45.

B. Reichstein, On expressing a cubic form as a sum of cubes of linear forms, Linear Algebra Appl., 86 (1987), 91–122, MR0870934 (88e:11022).

46.

B. Reichstein, On Waring’s problem for cubic forms, Linear Algebra Appl., 160 (1992), 1–61, MR1137842 (93b:11048).

47.

B. Reznick, Sums of even powers of real linear forms, Mem. Amer. Math. Soc., 96 (1992), no. 463, MR1096187 (93h:11043).

48.

B. Reznick, Uniform denominators in Hilbert’s Seventeenth Problem, Math. Z., 220 (1995), 75–97, MR1347159 (96e:11056).

49.

B. Reznick, Homogeneous polynomial solutions to constant coefficient PDE’s, Adv. Math., 117 (1996), 179–192, MR1371648 (97a:12006).

50.

B. Reznick, Laws of inertia in higher degree binary forms, Proc. Amer. Math. Soc., 138 (2010), 815–826, arXiv:0906.5559, MR26566547 (2011e:11074).

51.

B. Reznick, Blenders, in Notions of Positivity and the Geometry of Polynomials (P. Branden, M. Passare, M. Putinar, editors), Trends in Math., Birkhuser, Basel, 2011, pp. 345–373, arXiv:1008.4533v1.

52.

G. Salmon, Lesson introductory to the modern higher algebra, fifth edition, Chelsea, New York, 1964.

53.

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.

54.

J. J. Sylvester, On a remarkable discovery in the theory of canonical forms and of hyperdeterminants, originally in Phiosophical Magazine, vol. 2, 1851; Paper 42 in Mathematical Papers, Vol. 1, Chelsea, New York, 1973. Originally published by Cambridge University Press in 1904.

55.

J. J. Sylvester, On an elementary proof and demonstration 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.

56.

J.-C. Yakoubsohn, On Newton’s rule and Sylvester’s theorems, J. Pure Appl. Algebra, 65 (1990), 293–309, MR1072286 (91j:12002).

Title: On the Length of Binary Forms
Author: Bruce Reznick
Publisher: Springer New York
Book: Quadratic and Higher Degree Forms
Print ISBN: 978-1-4614-7487-6

Electronic ISBN: 978-1-4614-7488-3

Copyright Year: 2013
DOI: https://doi.org/10.1007/978-1-4614-7488-3_8

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