The 2025 ISSAC conference in Guanajuato, Mexico, marks the 50th event in this significant series, making it an ideal moment to reflect on the field’s history. This paper reviews the formative years of symbolic computation up to 1975, fifty years ago. By revisiting a period unfamiliar to most current participants, this survey aims to shed light on once-pressing issues that are now largely resolved and to highlight how some of today’s challenges were recognized earlier than expected.

We present a deterministic algorithm for deciding if a polynomial ideal, with coefficients in an algebraically closed field K of characteristic zero, of which we know just some very limited data, namely: the number n of variables, and some upper bound for the geometric degree of its zero set in Kn, is or not the zero ideal. The algorithm performs just a finite number of decisions to check wheather a point is or not in the zero set of the ideal. Moreover, we extend this technique to test, in the same fashion, if the elimination of some variables in the given ideal yields or not the zero ideal. Finally, the role of this technique in the context of automated theorem proving of elementary geometry statements, is presented, with references to recent documents describing the excellent performance of the already existing prototype version, implemented in GeoGebra.

This paper deals with a remarkable class of curves (in general r-space) that the two first authors have named "hypercircles" (see [2]). As shown there, such curves appear in the CAD context, when aiming towards finding a parametric representation with simpler coefficients (i.e. without algebraic numbers) for a given parametric curve. In fact, it turns out that the crucial point to solve the simplification problem in general is to solve this same problem for hypercircles [2]. Here we present an algorithm that, for a given parametrization of a hypercircle U, over an algebraic extension, namely, φ(t) ∈ K(α)(t)r, computes the linear fraction over K(α)(t) that generates this hypercircle (and, in particular, a parametrization of the curve over K).

Given a variety V, implicitly defined over an algebraic separable field extension k(alpha), A. Weil [5] developed a restriction technique (called by him a descente method),that associates to V a suitable k-variety W, such that many properties of V can be analyzed by merely looking at W, that is, by descending to the base field k. In this paper we present a parametric counterpart, for curves, of Weil's construction. As an application, we state some simple algorithmic criteria over the variety W that translate, for instance, the k-definability of a parametric curve V, or the existence of an infinite number of L-rational points in V.

Let K be a given computable field of characteristic zero and let ILbe a finite field extension of K, with algebraic closure F. Assume a rational parametrization P(t) E L(t)nof some irreducible curve C in the affine n-space over F is also given.In this paper we will show, first, how to decide -without imp]icitization algorithms-whether the given curve C is definable (by a set of equations with coefficients) over K; and, if this is the case, we will determine -without computing the implicit, equation set and then using parametrization techniques a reparametrizatiou of P(t) over the smallest possible field extension of K; that is, over a field extension of K of degree at most two.

When symbolically solving inverse kinematic problems for robot classes, we deal with computations on ideals representing these robot's geometry. Therefore, such ideals must be considered over a base field K, where the parameters of the class (and also the possible relations among them) are represented. In this framework we shall prove that the ideal corresponding to the general 6R manipulator is real and prime over K. The practical interest of our result is that it confirms that the usual inverse kinematic equations of this robot class do not add redundant solutions and that this ideal cannot be “factorized”, establishing therefore, Kova´cs [7] conjecture. We prove also that this root class has six degrees of freedom (i.e. the corresponding ideal is six-dimensional), even over the extended field K, which is the algebraic counterpart to the fact that the 6R manipulator is completely general. Our proof uses, as intermediate step, some dimensionality analysis of the Elbow manipulator, which is a specialization of the 6R.

Voronoi diagrams in the plane for strictly convex distances have been studied in [3], [5] and [7]. These distances induce the usual topology in the plane and, moreover, the Voronoi diagrams they produce enjoy many of the good properties of Euclidean Voronoi diagrams. Nevertheless, we show (Th.1) that it is not possible to transform, by means of a bijection from the plane into itself, the computation of such Voronoi diagrams to the computation of Euclidean Voronoi diagrams (except in the trivial case of the distance being affinely equivalent to the Euclidean distance). The same applies if we want to compute just the topological shape of a Voronoi diagram of at least four points (Th.2).

The inverse kinematics problem asks to determine the configurations of a robot manipulator that places the tip of the robot in a given position and orientation. The generalization of this problem, called path tracking or path following, is to determine if there exists a path of a manipulator in which the tip follows a given path. We study two different versions of this generalized problem: the tracking of the image of a path and the tracking of a path, giving to both of them an algorithmic answer. The main tool to obtain this solution is to construct a Cylindrical Algebraic Decomposition (CAD) of the set of admissible placements of the manipulator, such that is follows the projection onto the variables of the tip and is compatible with a triangulation of the image of the given path. Testing some properties in an oriented graph built from the adjacency graph of this CAD, provides the solution to the mentioned problems.*

Article A practical implementation of two rational function decomposition algorithms Share on Authors: Jaime Gutierrez View Profile , Tomas Recio View Profile Authors Info & Claims ISSAC '92: Papers from the international symposium on Symbolic and algebraic computationAugust 1992 Pages 152–157https://doi.org/10.1145/143242.143298Online:01 August 1992Publication History 1citation264DownloadsMetricsTotal Citations1Total Downloads264Last 12 Months0Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Article Sturm-Habicht sequence Share on Authors: L. Gonzalez View Profile , H. Lombardi View Profile , T. Recio View Profile , M.-F. Roy View Profile Authors Info & Claims ISSAC '89: Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computationJuly 1989 Pages 136–146https://doi.org/10.1145/74540.74558Published:17 July 1989 36citation499DownloadsMetricsTotal Citations36Total Downloads499Last 12 Months11Last 6 weeks2 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access