Minimal resultant systems (Q5917724)

These are results the author proved more than a decade ago in his thesis. Let \(f_1,f_2, \dots, f_s\) be \(s\) forms in two variables with indeterminate coefficients and of some fixed degrees, \(d_1=\deg f_1, d_2=\deg f_2, \dots, d_s=\deg f_s\). One can construct a bunch of polynomials (indeed an ideal) in the coefficients of the \(f_i\)'s called a resultant system such that \(f_i\)'s have a common zero, for certain coefficients in a field, if and only if the resultant system has a zero at the point corresponding to these coefficients in the appropriate affine space of dimension \(=s+\sum d_i\). The question the author deals with here is to estimate the least number of polynomials in a resultant system. That this number is at least \(s-1\) is easy to see. And in fact for \(s=2\), we get the usual resultant of two polynomials as a resultant system. So let us assume that \(s \geq 3\). The author proves the following: Assume \(d_1 \geq d_2 \geq \cdots \geq d_s\) and \(d_1\neq d_3\). Then (a) Every resultant system contains at least \(N_s=s-1+ \sum^{i=s}_{i=3} d_i\) polynomials. (b) If \(s=3\), there exists a resultant system consisting of precisely \(N_3\) elements.