Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case

The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo \cite{gihemorpar} can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in \cite{gihemorpar} yields a method for symbolically solving a zero-dimensional polynomial equation system in the affine (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straight-line programs. The algorithm solves any affine zero-dimensional equation system in non-uniform sequential time that is polynomial in the length of the input description and an adequately defined {\em affine degree} of the equation system. Replacing the affine degree of the equation system by a suitably defined {\em real degree} of certain polar varieties associated to the input equation, which describes the hypersurface under consideration, and using straight-line program codification of the input and intermediate results, we obtain a method for the problem introduced above that is polynomial in the input length and the real degree.Comment: Late

On the intrinsic complexity of the arithmetic Nullstellensatz

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.Facultad de Ciencias Exacta

On the intrinsic complexity of the arithmetic Nullstellensatz

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.Facultad de Ciencias Exacta