J. R. Sendra

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

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 McCallums 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.

An algebraic analysis of conchoids to algebraic curves

We study the conchoid to an algebraic affine plane curve

Partial degree formulae for plane offset curves

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Hankel Matrices and Polynomials

from the perspective of algebraic geometry, analyzing their main algebraic properties. Beside

Hankel matrices and computer algebra

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Finite piecewise polynomial parametrization of plane rational algebraic curves

, the notion of conchoid involves a point A in the affine plane (the focus) and a non-zero field element d (the distance). We introduce the formal definition of conchoid by means of incidence diagrams. We prove that the conchoid is a 1-dimensional algebraic set having at most two irreducible components. Moreover, with the exception of circles centered at the focus A and taking d as its radius, all components of the corresponding conchoid have dimension 1. In addition, we introduce the notions of special and simple components of a conchoid. Furthermore we state that, with the exception of lines passing through A, the conchoid always has at least one simple component and that, for almost every distance, all the components of the conchoid are simple. We state that, in the reducible case, simple conchoid components are birationally equivalent to