ON THE SIMILARITY GROUP OF FORMS OF HIGHER DEGREE

[摘要] Let f = f(X) = f(X(1),..., X(n),) E k[X(1),...,X(n)] be a non-singular form of degree d greater than or equal to 3 over the field k (where char(k)= 0 or char(k) > d). The similarity group of f, S-k(f) is the subgroup of GL(n)(k) consisting of all matrices A such that f(AX) = lambda(A)f(X) for some lambda(A) is an element of k(X). Let Aut(k)(f) denote the automorphism group of f. Identifying k(X) with the group of scalar matrices in S-k(f), we show using Kummer's theory of fields that S-k(f)/Aut(k)(f)k(X) is a finite abelian group of exponent d. We apply this result to give a complete characterization of the similarity group of rational binaries of odd degree. (C) 1994 Academic Press, Inc.

[发布日期] 1994-09-15