Singular case of pairs of bilinear, quadratic, or Hermitian forms

The main object of this paper is a treatment of the equivalence of pairs of büinear forms in the singular case by a purely rational method.The problem was first discussed by Kronecker,f who employed the irrational canonical form due to Weierstrass for the auxüiary non-singular case, instead of the rational canonical form (13) employed here.It is then a simple matter to deduce in Parts II and III the criteria for the equivalence of pairs of symmetric or Hermitian büinear forms, or quadratic or Hermitian forms, in the singular case. I. Pairs or bilinear eorms in the singular caseLet xp and yp be büinear forms in the same variables Xi, • • • ,xT, yi} • • ■, y, with coefficients in a field F, such that xp and yp are not equivalent in F to forms both of which involve fewer than r variables Xi or fewer than s variables y¿.We shall treat the singular case in which either r ?¿s, or else r=s and the determinant of f=u s without loss of generality, since we may interchange the letters x and y.Write /,-, <pi, ypi for the partial derivatives of /, xp, yp with respect to *<.The Unear functions fi, • • ■ ,fr of yi, • • • , y. are Unearly dependent.Hence there exist homogeneous polynomials A < in u and v of the same degree with coefficients in F, such that the A ¿ are not all zero identically and such that £ Aifi=0 identicaUy in u, v, yu ■ • • , y.. Lemma 1.The degree of Ai, • • • , Ar in u and v is not changed if we apply to f non-singular linear transformations on the x's and the y's separately whose coefficients are independent of u and v.