Complexity Lower Bounds For Computation Trees With Elementary Transcendental Function Gates

. We consider computation trees which admit as gate functions along with the usual arithmetic operations also algebraic or transcendental functions like exp, log, sin, square root (defined in the relevant domains) or much more general, Pfaffian functions. A new method for proving lower bounds on the depth of these trees is developed which allows to prove a lower bound\Omega\Gamma p log N) for testing membership to a convex polyhedron with N facets of all dimensions, provided that N is large enough. I. Pfaffian Computation Trees. Definition 1. By a Pfaffian computation tree T we mean a generalization of an algebraic decision tree (see e.g. [1, 4, 12, 28, 29, 30]) in which at any node v of T a Pfaffian function f v in the variables X 1 ; : : : ; Xn (see the definition A2 in the Appendix) is attached, which satisfies the following properties. Let f v0 ; : : : ; f v ` ; f v `+1 = f v be the functions attached to all the nodes along the branch T v of T leading from the root v 0..

Complexity bounds for cylindrical cell decompositions of sub-Pfaffian sets

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