Séminaire Lotharingien de Combinatoire, B36a (1996), 18pp.

Peter Bürgisser and Michael Clausen

Algebraische Komplexitätstheorie I: Eine Einführung

Abstract. Algebraic complexity theory is the study of the intrinsic algorithmic difficulty of algebraic problems within an algebraic model of computation. Motivated by questions of numerical and symbolic computation, this branch of research originated in 1954 when Ostrowski inquired about the optimality of Horner's rule. We survey some of the techniques for proving lower bounds in the straight-line and computation tree model. Among these are Strassen's degree bound, which relies on the concept of the degree of an affine variety, as well as lower bounds in terms of the number of connected components (or higher Betti numbers) of semi-algebraic sets. As a particular application of these methods we discuss the optimality of the Knuth-Schönhage algorithm for computing the GCD of two univariate polynomials.

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