A delineability-based method for computing critical sets of algebraic surfaces

Authors:
Juan Gerardo Alcazar;Josef Schicho;Juan Rafael Sendra
Affiliations:
Departamento de Matemáticas, Universidad de Alcalá, E-28871-Madrid, Spain;RICAM, Austrian Academy of Sciences, A-4040 Linz, Austria;Departamento de Matemáticas, Universidad de Alcalá, E-28871-Madrid, Spain
Venue:
Journal of Symbolic Computation
Year:
2007

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Abstract

In this paper, we address the problem of determining a real finite set of z-values where the topology type of the level curves of a (maybe singular) algebraic surface may change. We use as a fundamental and crucial tool McCallum's theorem on analytic delineability of polynomials (see [McCallum, S., 1998. An improved projection operation for cylindrical algebraic decomposition. In: Caviness, B.F., Johnson, J.R. (Eds.), Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer Verlag, pp. 242-268]). Our results allow to algorithmically compute this finite set by analyzing the real roots of a univariate polynomial; namely, the double discriminant of the implicit equation of the surface. As a consequence, an application to offsets is shown.