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Abstract. Let K be a field and F1 ..... F,. homogeneous polynomials in the indeterminates X o ..... X, with coefficients in K. We describe an efficiently parallelizable single exponential time algorithm which computes the Chow form of the ideal I : = (F 1 ..... F,,), provided that I is unmixed. This algorithm requires only linear algebra computations over K. Let g2 be an algebraically closed field and V a closed algebraic subvariety of the n-dimensional projective space P" over .Q. Assume that V is equidimensional, i.e. assume that all the irreducible components of V have the same dimension dim (V). Then the intersection of V with a "generic" linear subvariety E of P" of codimension codim (E):= n -dim (E) is:In fact, V c~ E is not empty whenever E is a linear subvariety, generic or not, such that dim (V) > codim (E). Moreover this property may be taken as the definition of dimension of algebraic varieties. On the other hand, the number of points in the intersection of V with a "generic" linear subvariety E such that dim (V) = codim (E) does not depend on E; it is known as the degree of V and plays a central role in intersection theory. We write deg(V) for the degree of V.Let r:= d i m ( V ) + 1. Translating linear subvarieties of P" to homogeneous linear equation systems, we may interpret the set of all linear subvarieties E of P" with codim (E) = r as a quasiprojective variety whose projective closure is the set of linear subvarieties with codimension < r. The set consisting of those linear subvarieties which have non-empty intersection with V is a hypersurface of this

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