Rend. Sem. Mat. Univ. Poi. Torino Voi. 52,4(1994) - Seminario ...

358 A.K.Bhandari - N.SankaranProof It follows troni [1, corollary in § 6] that C(t) is the field of invariante ofG and is purely dillerentially transccndental over C. Thus conditions (i) and (iii) of thedefinitimi are satisfied in view of the above proposition.For condition (ii) suppose lliat C'(£i,...,t n ,ni, • • •,^n) is'a PVE of C(f\,..., t n )for linearly independent zeros TTI, ..., 7f n of /(/i,... ,ì n , y). Then t n — W(7fi,..., 7r n )^ 0. If overC(*i,.. .,*„>, then t u = a(* n ) = del(cy)* n , so that det(c i;/ ) == 1. Also it. follows from [1,Theorem 4 in jj 8] that (c i:j is an ortliogonal matrix. Thus f(t,y) = 0 is a generic equationfor the group G.Acknowledgment. The aulhors wish to thank the reteree for his suggestions.REFERENCES[1] GOLDMAN L., Special Ì7.ation and Picard-Vessioi f . lìwory, Tram. Amer. Matti. Soc. 85(1957),327-356.[2] KAPLANSKV l.,An Introdactionto Differential Algebra, lliimrdim, Paris, 1957.[3] KoLClllN E.R., Algebraic matrix groups and the Picard-Vessiot theoty of homogeneous linearordinary differentiall equations, Armals of Math. 49 (1948), 1-41.[4] NOETHER E., Gleiclumgen mit vorgeschriebener Grappe, Math. Annalen 78 (1910), 221-229.[5] SMITH G.W., Generic cyclicpolynomials ofodd degree, Communications in Algebra 19 (1991),3367-3391.Ashwani K. Bhandari - N. SankaranDepartment of Mathematics, Panjab UniversityChandigarh - 160014, India.Lavoro pen>enufo in redazione il 25.3.1993 e, informa definitiva, il 20.2.1994.