An algebraic analysis of conchoids to algebraic curves

We study the conchoid to an algebraic affine plane curve from the perspective of algebraic geometry, analyzing their main algebraic properties. Beside , 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 , and we show how special components can be used to decide whether a given algebraic curve is the conchoid of another curve. J. R. Sendra and J. Sendra supported by the Spanish “Ministerio de Educación y Ciencia” under the Project MTM2005-08690-C02-01.     

On the parameterization of algebraic curves

In this paper, by using the concept of resolvents of a prime ideal introduced by Ritt, we give methods for constructing a hypersurface which is birational to a given irreducible variety and birational transformations between the hypersurface and the variety. In the case of algebraic curves, this implies that for an irreducible algebraic curveC, we can construct a plane curve which is birational toC. We also present a method to find rational parametric equations for a plane curve if it exists. Hence we have a complete method of parameterization for rational algebraic curves.The work reported here was supported in part by the NSF Grants CCR-8702108 and 9117870On leave from Institute of Systems Science, Academia Sinica, Beijing 100080, P.R. China     

Rational parametrization of conchoids to algebraic curves

We study the rationality of each of the components of the conchoid to an irreducible algebraic affine plane curve, excluding the trivial cases of the lines through the focus and the circle centered at the focus and radius the distance involved in the conchoid. We prove that conchoids having all their components rational can only be generated by rational curves. Moreover, we show that reducible conchoids to rational curves have always their two components rational. In addition, we prove that the rationality of the conchoid component, to a rational curve, does depend on the base curve and on the focus but not on the distance. As a consequence, we provide an algorithm that analyzes the rationality of all the components of the conchoid and, in the affirmative case, parametrizes them. The algorithm only uses a proper parametrization of the base curve and the focus and, hence, does not require the previous computation of the conchoid. As a corollary, we show that the conchoid to the irreducible conics, with conchoid-focus on the conic, are rational and we give parametrizations. In particular we parametrize the Limaçons of Pascal. We also parametrize the conchoids of Nicomedes. Finally, we show how to find the foci from where the conchoid is rational or with two rational components.     

Special issue on silicon-germanium alloys

Editorial

Special issue on silicon-germanium alloys     

Group codes on certain algebraic curves with many rational points

We construct a series of algebraic geometric codes using a class of curves which have many rational points. We obtain codes of lengthq ² over _q , whereq = 2q ₀ ² andq ₀ = 2 ⁿ , such that dimension + minimal distance q ² + 1 – q ₀ (q – 1). The codes are ideals in the group algebra _q [S], whereS is a Sylow-2-subgroup of orderq ² of the Suzuki-group of orderq ² (q ² + 1)(q – 1). The curves used for construction have in relation to their genera the maximal number of GF _q -rational points. This maximal number is determined by the explicit formulas of Weil and is effectively smaller than the Hasse—Weil bound.Supported by Deutsche Forschungsgemeinschaft while visiting Essen University     

Parametrization of algebraic curves defined by sparse equations

We present a new method for the rational parametrization of plane algebraic curves. The algorithm exploits the shape of the Newton polygon of the defining implicit equation and is based on methods of toric geometry. The authors were supported by the FWF (Austrian Science Fund) in the frame of the research projects SFB 1303 and P15551.