In this paper we describe a new technique for obtaining lower bounds on restriced classes of nonmonotone arithmetic circuits. The heart of this technique is a complexity measure for multivariate polynomials, based on the linear span of their partial derivatives. We use the technique to obtain new lower bounds for computing symmetric polynomials and iterated matrix products.

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We present a logical approach to complexity over the real numbers with respect to the model of Blum, Shub and Smale. The logics under consideration are interpreted over a special class of two-sorted structures, called R-structures: They consist of a finite structure together with the ordered field of reals and a finite set of functions from the finite structure into R. They are a special case of the metafinite structures introduced recently by Grädel and Gurevich. We argue that R-structures provide the right class of structures to develop a descriptive complexity theory over R. We substantiate this claim by a number of results that relate logical definability on R-structures with complexity of computations of BSS-machines.

We present a new method to obtain lower bounds for the time complexity of polynomial evaluation procedures. Time, denoted by L, is measured in terms of nonscalar arithmetic operations. In contrast with known methods for proving lower complexity bounds, our method is purely combinatorial and does not require powerful tools from algebraic or diophantine geometry.