Evaluation Techniques for Zero-dimensional Primary Decomposition Extended Abstract

In this talk, we will present an algorithm that computes the local algebra of the roots of a zero-dimensional polynomial equations system, whose cost is polynomial in the number of variables, in the evaluation cost of the equations and in the Bézout number of the input system. Let K be a field of characteristic zero, and let f1, . . . , fm, g be polynomials in K[x1, . . . , xn] such that the system f1 = · · · = fm = 0 with g 6= 0 has only a finite set of solutions over the algebraic closure K̄ of K. We give an algorithm that computes the roots of the system together with the structure of their multiplicities. More precisely, our purpose is to compute the primary decomposition of the zerodimensional ideal (f1, . . . , fm) : g , where for any ideal J in K[x1, . . . , xn], J : g ∞ denotes the saturation of J with respect to g, that is, the ideal {f | ∃n ≥ 0, gf ∈ J }. Main Result. For any root p = (p1, . . . , pn) ∈ K̄ n of (f1, . . . , fm) : g , let K(p)[[x1 − p1, . . . , xn − pn]] denote the ring of formal series in x1 − p1, . . . , xn − pn over the algebraic field extension K(p) = K(p1, . . . , pn) of K. The local algebra of p as a root of (f1, . . . , fn) : g ∞ is the K(p)-algebra Dp = K(p)[[x1 − p1, . . . , xn − pn]]/((f1, . . . , fm) : g )p, where for any ideal J in K[x1, . . . , xn], the notation Jp stands for the ideal J extended to K(p)[[x1 − p1, . . . , xn − pn]]. The multiplicity μp of p is the dimension of the K(p)-algebra Dp. The matrices of the morphism of multiplication by the variables in some basis of Dp allow all computations in Dp. Here we propose a new algorithm for computing such matrices. The improvement of the exponents involved in the complexity is still in progress. Theorem 1 ([3]). Let K be a field of characteristic zero and let f1, . . . , fm, g be polynomials in K[x1, . . . , xn] of degree at most d, given by straight-line programs of size at most L. Let us assume that the system f1 = · · · = fm with g 6= 0 has only a finite set of solutions over the algebraic closure K̄ of K. The roots of the system together with the matrices of multiplication by the variables in some basis of their local algebra can be computed with a number of arithmetic operations in K which is polynomial in n, L and d. There is a probabilistic algorithm performing these computations. Its probability of returning the correct result relies on choices of elements in K. Choices for which the result is not correct are enclosed in strict