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arXiv · alg-geom/9609003

Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case

B. Bank·M. Giusti·J. Heintz·R. Mandel·G. M. Mbakop
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Abstract

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.

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B. Bank, M. Giusti, J. Heintz, R. Mandel, G. M. Mbakop. 1996-09-06. Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case. https://arxiv.org/abs/alg-geom/9609003

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