Subresultants and Generic Monomial Bases

Given n polynomials in n variables of respective degrees d_1,...,d_n, and a set of monomials of cardinality d_1...d_n, we give an explicit subresultant-based polynomial expression in the coefficients of the input polynomials whose non-vanishing is a necessary and sufficient condition for this set of monomials to be a basis of the ring of polynomials in n variables modulo the ideal generated by the system of polynomials. This approach allows us to clarify the algorithms for the Bezout construction of the resultant.Comment: 22 pages, uses elsart.cls. Revised version accepted for publication in the Journal of Symbolic Computatio

Effective Differential Nullstellensatz for Ordinary DAE Systems with Constant Coefficients

We give upper bounds for the differential Nullstellensatz in the case of ordinary systems of differential algebraic equations over any field of constants $K$ of characteristic $0$. Let $\vec{x}$ be a set of $n$ differential variables, $\vec{f}$ a finite family of differential polynomials in the ring $K\{\vec{x}\}$ and $f\in K\{\vec{x}\}$ another polynomial which vanishes at every solution of the differential equation system $\vec{f}=0$ in any differentially closed field containing $K$. Let $d:=\max\{\deg(\vec{f}), \deg(f)\}$ and $\epsilon:=\max\{2,{\rm{ord}}(\vec{f}), {\rm{ord}}(f)\}$. We show that $f^M$ belongs to the algebraic ideal generated by the successive derivatives of $\vec{f}$ of order at most $L = (n\epsilon d)^{2^{c(n\epsilon)^3}}$, for a suitable universal constant $c>0$, and $M=d^{n(\epsilon +L+1)}$. The previously known bounds for $L$ and $M$ are not elementary recursive