Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 642.10021
Autor: Dixmier, Jacques; Erdös, Paul; Nicolas, Jean-Louis
Title: Sur le nombre d'invariants fondamentaux des formes binaires. (On the number of fundamental invariants of binary forms.) (In French)
Source: C. R. Acad. Sci., Paris, Sér. I 305, 319-322 (1987).
Review: For an arbitrary positive integer d, let \omegad denote the number of fundamental invariants for binary forms of degree d. For odd d \geq 3, V. G. Kac [Lect. Notes Math. 996, 74-108 (1983; Zbl 534.14004)] minorized \omegad in the following way. For an arbitrary positive integer n, consider the congruence x1+2x2+...+(n-1)xn- 1\equiv 0 (mod n) where x1,...,xn-1 are nonnegative integers, not all 0. A solution is said to be indecomposable if it is not the sum of two solutions. Denoting the number of solutions indecomposable by F(n), V.G.Kac [ibid.] proved that (1) \omegad \geq F(d-2) for odd d \geq 3. Let p(n) denote the number of unrestricted partitions of n (and \phi Euler's totient function). (1) immediately implies that
(2) \omegad \geq p(d-2)+\phi(d-2)-2 for odd d \geq 3. In the paper under review, the authors improve the lower bounds of F(n) and obtain that
(3) \omegad >> p(d)· d ½(log d log log d)-1 for odd d. As to the even d's, using two results of type (1), proved by V. L. Popov [Izv. Akad. Nauk SSSR, Ser. Mat. 47, No.3, 544-622 (1983; Zbl 573.14003); J. Reine Angew. Math. 341, 157-173 (1983; Zbl 525.14007)], the authors obtain that
\omegad >> p(d/4)· d ½(log d log log d)-1 for d\equiv 0 (mod 4) and with \epsilon > 0
\omegad >> \epsilonp(d/2)· \exp {(\pi 3-- \epsilon)d ½ log log d (log d)-1} for d\equiv 2 (mod 4).
Reviewer: M.Szalay
Classif.: * 11E76 Forms of degree higher than two
15A72 Vector and tensor algebra
11P81 Elementary theory of partitions
Keywords: binary forms of higher degree; fundamental invariants; partitions
Citations: Zbl 534.14004; Zbl 573.14003; Zbl 525.14007
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