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Let S 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f 1 , . . . , f p . This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S 0 . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of S 0 . This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f 1 , . . . , f p and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system f 1 , . . . , f p .

We present a new method for solving symbolically zero-dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of problem adapted data structures : arithmetic networks and straight-line programs. For sequential time complexity measured by network size we obtain the following result : it is possible to solve any affine or toric zero-dimensional equation system in non-uniform sequential time which is polynomial in the length of the input description and the "geometric degree" of the equation system. Here, the input is thought to be given by a straight-line program (or alternatively in sparse representation), and the length of the input is measured by number of variables, degree of equations and size of the program (or sparsity of the equations). The geometric degree of the input system has to be adequately defined. It is always bounded by the algebraic-combinatoric "Bézout number" of the system which is given by the Hilbert function of a suitable homogeneous ideal. However, in many important cases, the value of the geometric degree of the system is much smaller than its Bézout number since this geometric degree does not take into account multiplicities or degrees of extraneous components (which may appear at infinity in the affine case or may be contained in some coordinate hyperplane in the toric case).Our method contains a new application of a classic tool to symbolic computation : we use Newton iteration in order to simplify straight-line programs occurring in elimination procedures. Our new technique allows for practical implementations a meaningful characterization of the intrinsic algebraic complexity of typic elimination problems and reduces the still unanswered question of their intrinsic bit complexity to algorithmic arithmetics. However our algorithms are not rational anymore as are the usual ones in elimination theory. They require some restricted computing with algebraic numbers. This is due to its numeric ingredients (Newton iteration). Nevertheless, at least in the case of polynomial equation systems depending on parameters, the practical advantage of our method with respect to more traditional ones in symbolic and numeric computation is clearly visible. Our approach produces immediately a series of division theorems (effective Nullstellensätze) with new and more differentiated degree and complexity bounds (we shall state two of them).It should be well understood that our method does not improve the well known worst-case complexity bounds for zero-dimensional equation solving in symbolic and numeric computing.Part of the results of this paper were announced in [25].

We exploit the fact that the set of all polynomials P~ ~[Xl,..

Let V be a closed algebraic subvariety of the n-dimensional projective space over the complex or real numbers and suppose that V is non-empty and equidimensional. The classic notion of a polar variety of V associated with a given linear subvariety of the ambient space of V was generalized and motivated in Bank et al. (Kybernetika 40 (2004), to appear). As particular instances of this notion of a generalized polar variety one reobtains the classic one and an alternative type of a polar variety, called dual. As main result of the present paper we show that for a generic choice of their parameters the generalized polar varieties of V are empty or equidimensional and smooth in any regular point of V. In the case that the variety V is affine and smooth and has a complete intersection ideal of definition, we are able, for a generic parameter choice, to describe locally the generalized polar varieties of V by explicit equations. Finally, we indicate how this description may be used in order to design in the context of algorithmic elimination theory a highly efficient, probabilistic elimination procedure for the following task: In case, that the variety V is Q-definable and affine, having a complete intersection ଁ ideal of definition, and that the real trace of V is non-empty and smooth, find for each connected component of the real trace of V an algebraic sample point.

We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. ¿From this algorithm, we derive a new procedure for the decision of consistency of polynomial equation systems whose bit complexity is subexponential, too. As a byproduct, we analyze the division of a polynomial modulo a reduced complete intersection ideal and from this, we obtain an intrinsic lower bound for the logarithmic height of diophantine approximations to a given solution of a zero-dimensional polynomial equation system. This result represents a multivariate version of Liouville's classical theorem on approximation of algebraic numbers by rationals. A special feature of our procedures is their polynomial character with respect to the mentioned geometric invariants when instead of bit operations only arithmetic operations are counted at unit cost. Technically our paper relies on the use of straight-line programs as a data structure for the encoding of polynomials, on a new symbolic application of Newton's algorithm to the Implicit Function Theorem and on a special, basis independent trace formula for affine Gorenstein algebras.

DEDICATED TO SHMUEL WINOGRADIn this paper we apply for the first time a new method for multivariate equation solving which was developed for complex root determination to the real case. Our main result concerns the problem of finding at least one representative point for each connected component of a real compact and smooth hypersurface. The basic algorithm yields a new method for symbolically solving zero-dimensional polynomial equation systems over the complex numbers. One feature of central importance of this algorithm is the use of a problem-adapted data type represented by the data structures arithmetic network and straight-line program (arithmetic circuit). The algorithm finds the complex solutions