Mathematics Subject Classification (1991): 12D10, 26D10, 30C15, 65H10, 65H15
Springer Online Journal Archives 1860-2000
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  and  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.
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