On some resultant identities (Q583327)

Let \(p(t_ 1,t_ 2)=\sum^{n}_{i=0}a_ it^ i_ 1t_ 2^{n-i}\) and \(q(t_ 1,t_ 2)=\sum^{m}_{i=0}b_ it^ i_ 1t_ 2^{m-i}\) be homogeneous polynomials over an integral domain K. The resultant R(p,q) is the determinant of the Sylvester matrix S(p,q). If \(\phi,\psi\) are polynomials let \(\tilde p=p(\phi,\psi)\) and \(\tilde q=q(\phi,\psi)\). The authors prove that there exist two ``resultant-like'' matrices, \(C_ 1\) and \(C_ 2\) so that \(C_ 1\bullet S(\tilde p,\tilde q)=[S(p,q)\otimes I_ r]\bullet C_ 2\), where \(r=\max (\deg \phi,\deg \psi)\). The matrices \(C_ 1\) and \(C_ 2\) depend only on \(\phi\) and \(\psi\) and are invertible if and only if \(\phi\) and \(\psi\) are coprime polynomials. They also give a proof of a result of \textit{U. Helmke} on Bézoutians which says that there exists a matrix C such that \(B(\tilde p,\tilde q)=C'[B(p,q)\otimes B(\phi,\psi)]C\) and they give an explicit expression for C which turns out to be a resultantlike matrix depending only on \(\phi\) and \(\psi\).