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 an alternative data structure: arithmetic networks and straight-line programs with FOR gates. For sequential time complexity measured by the size of these networks 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). Geometric degree 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 is much smaller than the Bézout number since it does not take into account multiplicities or degrees of extraneous components (which are at infinity in the affine case or contained in some coordinate hyperplane in the toric case). Finally, we announce the result that FOR gates can be avoided by a method which, based on Newton iteration, pulls back the whole question to ordinary arithmetic networks and straight-line programs. In this context, our complexity bounds remain valid. However, this second procedure is not rational anymore because it requires 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. 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.

These pages are a first attempt to compare the efficiency of symbolic and numerical analysis procedures that solve systems of multivariate polynomial equations. In particular, we compare Kronecker's solution (from the symbolic approach) with approximate zero theory (introduced by S. Smale as a foundation of numerical analysis). For this purpose we show upper and lower bounds of the bit length of approximate zeros. We also introduce efficient procedures that transform local Kronecker solutions into approximate zeros and conversely. As an application of our study we exhibit an efficient procedure to compute splitting field and Lagrange resolvent of univariate polynomial equations. We remark that this procedure is obtained by a convenient combination of both approaches (numeric and symbolic) to multivariate polynomial solving.

We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorithmic procedure computing the polynomials and constants occurring in a Bézout 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 polynomial 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 πS(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 characteristic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.

In this extended abstract we deal with the relations between the numerical/diophantine approximation and the symbolic/algebraic geometry approachs to solving of multivariate diophentine polynomial systems, obtaining several consecuences ranging from diophantine approximation to effective number theory.

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 : GAGE, Centre de Mathématiques. École Polytechnique. F-91228, Palaiseau Cedex. FRANCE : Dept. de Matemáticas, Estad́ıstica y Computación. F. de Ciencias. U. Cantabria. E-39071 SANTANDER, Spain : Research was partially supported by the following European, French and Spanish grants : PoSSo ESPRIT/BRA 6846, GDR CNRS 1026 MEDICIS, DGCYT PB92–0498–C02–01 and PB93–0472–C02–02

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 nonuniform 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 "Bezout 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 Bezout 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 tonic 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 Nullstellensatze) 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]. (C) 1998 Elsevier Science B.V.

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.

Solving elimination problems requires only a moderate amount of bit operations if one uses appropriate data structures (as e.g. straight-line programs) for the encoding of polynomials. We present an elimination procedure whose bit complexity is polynomial in the input size and in the value of two suitably defined invariants which reflect the geometric degree and the arithmetic height of the input system. From our complexity bound we deduce by means of an appropriate effective Nullstellensatz a multivariate and intrinsic version of Liouville's classical theorem on approximation of algebraic numbers by rationals. Consequences for practically solving systems of polynomial equations are drawn.

We show lower bounds for the parallel complexity of membership problems in semialgebraic sets. Our lower bounds are obtained from the Euler characteristic and the sum of Betti numbers. We remark that these lower bounds are polynomial (an square root) in the sequential lower bounds obtained by Andrew C.C. Yao.