Sur le nombre d`invariants fondamentaux des formes binaires

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MR909556 (89a:11040)
Dixmier, Jacques; Erdős, Paul (H-AOS); Nicolas, Jean-Louis (F-LIMO)
Sur le nombre d’invariants fondamentaux des formes binaires. (French. English summary)
[On the number of fundamental invariants of binary forms]
C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 8, 319–322.
11E76 (11N45 15A72)
References: 0
Reference Citations: 2
Review Citations: 0
Denote by Vd the vector space of homogeneous polynomials in two variables of degree d with
complex coefficients. Let C[Vd ] be the algebra of polynomial functions Vd → C. The group G =
SL(2, C) operates in a natural way on Vd and so on C[Vd ]. Let Ad be the subalgebra of G-invariant
elements in C[Vd ]. Denote by ωd the number of elements in any minimal generating system of the
algebra Ad (number of fundamental invariants of binary forms of degree d). In the present paper
some lower bounds for ωd obtained by V. G. Kac [in Invariant theory (Montecatini, 1982), 74–108,
Lecture Notes in Math., 996, Springer, Berlin, 1983; MR0718127
√ (85j:14088)] and V. L.−1Popov
are improved. Thus, for odd d → ∞ it is proved that lim inf ωd [ dp(d)/ log d log log d] > 0.
Here p(d) denotes the number of all partitions of d. Similar results are deduced for d ≡ 2 mod 4
or d ≡ 0 mod 4, respectively, as d → ∞.
Reviewed by O. H. Körner
c Copyright American Mathematical Society 1989, 2005