Distance bounds of ε-points on hypersurfaces

e-Points were introduced by the authors (see [S. Perez-Diaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic curves by lines, Theoret. Comput. Sci. 315(2-3) (2004) 627-650 (Special issue); S. Perez-Diaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic surfaces by lines, Comput. Aided Geom. Design 22(2) (2005) 147-181; S. Perez-Diaz, J.R. Sendra, J. Sendra, Distance properties of e-points on algebraic curves, in: Series Mathematics and Visualization, Computational Methods for Algebraic Spline Surfaces, Springer, Berlin, 2005, pp. 45-61]) as a generalization of the notion of approximate root of a univariate polynomial. The notion of e-point of an algebraic hypersurface is quite intuitive. It essentially consists in a point such that when substituted in the implicit equation of the hypersurface gives values of small module. Intuition says that an e-point of a hypersurface is a point close to it. In this paper, we formally analyze this assertion giving bounds of the distance of the e-point to the hypersurface. For this purpose, we introduce the notions of height, depth and weight of an e-point. The height and the depth control when the distance bounds are valid, while the weight is involved in the bounds.