Perrucci, Daniel; Roy, Marie-Françoise Zero-nonzero and real-nonreal sign determination. (English) Zbl 1309.14047 Linear Algebra Appl. 439, No. 10, 3016-3030 (2013). Summary: We consider first the zero-nonzero determination problem, which consists in determining the list of zero-nonzero conditions realized by a finite list of polynomials on a finite set \(Z \subset C^k\) with \(C\) an algebraic closed field. We describe an algorithm to solve the zero-nonzero determination problem and we perform its bit complexity analysis. This algorithm, which is in many ways an adaptation of the methods used to solve the more classical sign determination problem, presents also new ideas which can be used to improve sign determination. Then, we consider the real-nonreal sign determination problem, which deals with both the sign determination and the zero-nonzero determination problem. We describe an algorithm to solve the real-nonreal sign determination problem, we perform its bit complexity analysis and we discuss this problem in a parametric context. Cited in 1 Document MSC: 14P10 Semialgebraic sets and related spaces 14Q20 Effectivity, complexity and computational aspects of algebraic geometry Keywords:polynomial equations and inequations; systems; sign determination; complexity PDFBibTeX XMLCite \textit{D. Perrucci} and \textit{M.-F. Roy}, Linear Algebra Appl. 439, No. 10, 3016--3030 (2013; Zbl 1309.14047) Full Text: DOI arXiv References: [1] Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise, Algorithms in Real Algebraic Geometry, Algorithms Comput. Math., vol. 10 (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1102.14041 [2] Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise, Algorithms in real algebraic geometry, current online version, available at · Zbl 1102.14041 [3] Canny, John, Improved algorithms for sign determination and existential quantifier elimination, Comput. J., 36, 5, 409-418 (1993) · Zbl 0789.68079 [4] Lickteig, Thomas; Roy, Marie-Françoise, Sylvester-Habicht sequences and fast Cauchy index computations, J. Symb. Comput., 31, 3, 315-341 (2001) · Zbl 0976.65043 [5] Perrucci, Daniel, Linear solving for sign determination, Theor. Comput. Sci., 412, 35, 4715-4720 (2011) · Zbl 1221.68301 [7] Roy, Marie-Françoise; Szpirglas, Aviva, Complexity of computation on real algebraic numbers, J. Symb. Comput., 10, 1, 39-51 (1990) · Zbl 0723.68054 [8] von zur Gathen, Joachim; Gerhard, Jürgen, Modern Computer Algebra (1999), Cambridge University Press: Cambridge University Press New York · Zbl 0936.11069 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.