Subresultants and locally nilpotent derivations.
For \(P\), \(Q\) univariate polynomials of degree \(p\), resp. \(q\), over a commutative ring with unity, let \(\delta (p,q)\) denote \(\min (p,q)\) if \(p\neq q\) and \(p-1\) otherwise. For \(i\) at most \(\delta (p,q)\), let \(Sr_i (P,Q)\) denote the \(i\)th subresultant polynomial associated to \(P\) and \(Q\). It is proved that \(Sr_i (P,Q)\) is the determinant of a \((p+q-2i)\times (p+q-2i)\) matrix whose entries are taken from the list of coefficients of \(P\), \(P'\), \(\frac{P''}{2}, \ldots, \frac{P^{(p)}}{p!}\), \(Q\), \(Q'\), \(\frac{Q''}{2}, \ldots, \frac{Q^{(q)}}{q!}\). Moreover, the determinants of suitable \((p+q-2i-1)\times (p+q-2i-1)\) matrices provide polynomials \(U_i\) and \(V_i\) such that \(U_iP+ V_iQ=Sr_i (P,Q)\). These properties are established by specialization of similar results stated for Lie series associated to locally nilpotent derivations on commutative rings containing the rationals.
scientific article; zbMATH DE number 2062871
scientific article