Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems
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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  and  mainly for the case that \(\) 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.
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