Theory of multiple polynomial remainder sequence (Q1076728)

For a set of polynomials \(\{P_ 0^{(1)}(x),...,P_ 0^{(m)}(x)\}\) with coefficients in an integral domain I a multiple polynomial remainder sequence (multi-PRS in short) \(\{P_ i^{(1)}(x),...,P_ i^{(m)}(x)\}\), \(i=1,2,..\). is given by suitable formulas. It is proved that for any multi-PRS \(\{P_ i^{(1)},...,P_ i^{(m)}\}\), \(i=1,2,...\), there exist PRS-matrices \(M^{(\mu)}_{i,j}\), \(\mu =1,2,...,m\), \(j=0,1,...,i-1\), such that \(P_ i^{(\mu)}\sim | M^{(\mu)}_{i,j}|\) or \(P_ i^{(\mu)}=\lambda^{(\mu)}_{i,j}| M^{(\mu)}_{i,j}|\) where \(M^{(\mu)}_{i,j}\) is composed of \(P_ j^{(k)}(x)\), \(k=1,2,...,m\) and their coefficients. The main properties as well as the inverse-reduction of the matrices \(M^{(\mu)}_{i,j}\) are considered. On this base the factor \(\lambda^{(\mu)}_{i,0}\) is calculated and several algorithms for computing multi-PRS over I are presented.