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 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.

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We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorith mic procedure computing the polynomials and constants occurring in a Bezout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the first time height estimates of intrinsic type for the polynomials and constants appearing, again poly nomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function 7ls(x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteris tic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.

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