On Semi-Discriminants of Ternary Forms

O. E. Glenn
1911 Transactions of the American Mathematical Society  
Introduction. It is well known that the number of independent conditional relations which must exist among the coefficients of a ternary form of order m in order that it should be factorable into linear factors, distinct term for term, is Jm(m -1). Several writers, f among them Brill, and Gordan, have published methods for the determination of such sets of relations. Their results are, as a rule, expressed in the form of a covariant, the identical vanishing of which gives necessary and
more » ... essary and sufficient conditions for the factorability. These methods are somewhat indirect, and from certain standpoints are unsatisfactory for the additional reason that the set of conditions given by the identical vanishing of such a covariant is always redundant. Our aim in this paper has been to develop a direct method of attacking this problem. Our method leads to a set of conditional relations containing the exact minimum number \m (m -1 ) ; that is, it leads to a set of \m(m -1 ) independent semin variants of the form, whose simultaneous vanishing gives necessary and sufficient conditions for the factorability. We shall call these seminvariants semi-discriminants J of the form. They are all of the same degree 2m -1 ; and are readily formed for any order m as simultaneous invariants of a certain set of binary quantics related to the original ternary form. If a polynomial, f3m, of order m, and homogeneous in three variables (xx, x2, xs) is factorable into linear factors, its terms in (xx, x2) must furnish the (a;" x2) terms of those factors. Call these terms collectively a™x, and the terms linear in x3 collectively x3a"'~l. Then if the factors of the former were known, and were distinct, say
doi:10.2307/1988673 fatcat:5ayjmecw4fhu3iyinnvbfnlqmi