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00016646 Dickenstein, Alicia M. Sessa, Carmen Duality methods for the membership problem. Effective methods in algebraic geometry, Proc. Symp., Castiglioncello/Italy 1990, Prog. Math. 94, 89-103 (1991). 1991 EN membership problem of complex polynomial ideals complexity residual operators Zbl 0721.00009 [For the entire collection see Zbl 0721.00009.] Given an ideal $I$ in $\bbfC[z\sb 1,\ldots,z\sb n]$ generated by polynomials $f\sb 1,\ldots,f\sb s$ of degree less or equal than $d$, we propose effective criteria to decide membership in $I$ and in the radical of $I$, in the zero dimensional and complete intersection cases. --- In fact for any $k\in N$, we show how to construct a homogeneous system of linear equations $S(I)$ (resp. $S(\text{rad}(I)))$ such that the vector of coefficients of a given polynomial $P$ with $\deg(P)\le k$ is a solution of $S(I)$ (resp. $S(\text{rad}(I)))$ iff $P\in I$ (resp. $P\in\text{rad}(I))$. As a by-product, we are able to compute generators for $\text{rad}(I)$ in case $\dim(I)=0$, or if an a priori degree bound of type $d\sp{0(n)}$ is given in the case of complete intersection. All our computations have simply exponential in $n$ and polynomial in $d,k$ and $s$ sequential complexity bounds. Our methods are essentially based on the theory of residual operators in several complex variables, and extend the results of our paper in J. Pure Appl. Algebra 74, No. 2, 149-158 (1991)]. This is combined with admissible time computations of certain quotient ideals and with the ``Zariski-Samuel local duality'' for quotient ideals of irreducible ideals in a local ring. A.M.Dickenstein (Buenos Aires)