Computation of Primary Decomposition with the Zeros of an Ideal

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In this paper, we give a new approach to the computation of primary decomposition and associated prime components of a zero-dimensional polynomial ideal(f_1, f_2・・・, f_n), where f_i are multivariate polynomials on Z(the ring of integer). Over the past several years, a considerable number of studies have been made on the computation of primary decomposition of a zero-dimensional polynomial ideal. Many algorithms to compute primary decomposition are proposed. Most of the algorithms recently proposed are based on Groebner basis. However, the computation of Groebner basis can be very expensive to perform. Some computations are even impossible because of the physical limitation of memory in a computer. On the other hand, recent advance in numerical methods such as homotopy method made access to the zeros of a polynomial system relatively easy. Hence, instead of Groebner basis, we use the zeros of a given ideal to compute primary decomposition and associated prime components. More specifically, given a zerodimensional ideal, we use LLL reduction algorithm by Lenstra et al.to determine the integer coefficients of irreducible polynomials in the ideal. It is shown that primary decomposition and associated prime components of the ideal can be computed, provided the zeros of the ideal are computed with enough accuracy. A numerical experiment is given to show effectiveness of our algorithm.
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1998-04-25