Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems

Summary.

The eigenproblem method calculates the solutions of systems of polynomial equations $f_1(x_1, \ldots , x_s)=0,\ldots,f_m(x_1, \ldots , x_s)=0$. It consists in fixing a suitable polynomial $f$ and in considering the matrix $A_f$ corresponding to the mapping $[p] \mapsto [f\cdot p]$ where the equivalence classes are modulo the ideal generated by $f_1, \ldots , f_m.$ The eigenspaces contain vectors, from which all solutions of the system can be read off. This access was investigated in [1] and [16] mainly for the case that \( A_f is nonderogatory. In the present paper, we study the case where $f_1, \ldots , f_m$ have multiple zeros in common. We establish a kind of Jordan decomposition of $A_f$ reflecting the multiplicity structure, and describe the conditions under which $A_f$ is nonderogatory. The algorithmic analysis of the eigenproblem in the general case is indicated.