In this survey article we describe how symbolic computation in algebra and geometry leads to symbolic, i.e., formula solutions of algebraic differential equations. Symbolic solutions of algebraic differential equations can be derived from parametrizations of corresponding algebraic varieties. Such parametrizations in turn can be computed by elimination methods, i.e., methods for solving systems of polynomial equations.

In this paper, we provide an algorithm to compute explicit rational solutions of a rational system of autonomous ordinary differential equations (ODEs) from its rational invariant algebraic curves. The method is based on the proper rational parametrization of these curves and the fact that by linear reparametrizations, we can find the rational solutions of the given system of ODEs. Moreover, if the system has a rational first integral, we can decide whether it has a rational general solution and compute it in the affirmative case.

In this paper we study non-autonomous algebraic ODEs F(x,y,y′)=0, where F(x,y,z)∈Q¯[x,y,z], provided a proper rational parametrization P(s,t) of the corresponding algebraic surface F(x,y,z)=0. We show the relation between a rational general solution of the non-autonomous differential equation F(x,y,y′)=0 and a rational general solution of its associated autonomous system with respect to P(s,t). The degrees of a rational solution (s(x),t(x)) of the associated system are studied by giving a degree bound for t(x) in terms of the degree of s(x) and the degree with respect to s of the first component of P(s,t). We also give a criterion for the existence of rational general solutions of the associated system provided a degree bound of its rational general solutions. The criterion is based on the vanishing of the differential pseudo remainder of Gao’s differential polynomials with respect to the chain of differential polynomials derived from the associated system. We use this criterion to classify all autonomous linear systems of ODEs of order 1 having a rational general solution.

We extend the theory of Gröbner bases to difference-differential modules. The main goal of this paper is to present and verify algorithms for constructing Gröbner bases for such difference-differential modules. To this aim we introduce the concept of generalized term order on Nm × Zn and on difference-differential modules.

The degree of a rational map measures how often the map covers the image variety. In particular, when the rational map is a parametrization, the degree measures how often the parametrization traces the image. We show how the degree of rational maps between algebraic curves can be determined efficiently. In the process, we also give a complete proof of Sederberg's approach for making a parametrization proper.

A canal surface S, generated by a parametrized curve m(t), in R3 is the envelope of the set of spheres with radius r(t) centered at m(t). This concept generalizes the classical offsets (for r(t) = const) of plane curves. In this paper we develop elementary symbolic methods for generating a rational parametrization of canal surfaces generated by rational curves m(t) with rational radius variation r(t). In a pipe surface r(t) is constant.

In this paper, we study fundamental properties of real curves, especially of rational real curves, and we derive several algorithms to decide the reality and rationality of curves in the complex plane. Furthermore, if the curve is real and rational,

Article CASA: A computer algebra package for constructive algebraic geometry Share on Authors: R. Gebauer RISC-Linz, Johannes Kepler Universität, A-4040 Linz, Austria RISC-Linz, Johannes Kepler Universität, A-4040 Linz, AustriaView Profile , M. Kalkbrener RISC-Linz, Johannes Kepler Universität, A-4040 Linz, Austria RISC-Linz, Johannes Kepler Universität, A-4040 Linz, AustriaView Profile , B. Wall RISC-Linz, Johannes Kepler Universität, A-4040 Linz, Austria RISC-Linz, Johannes Kepler Universität, A-4040 Linz, AustriaView Profile , F. Winkler RISC-Linz, Johannes Kepler Universität, A-4040 Linz, Austria RISC-Linz, Johannes Kepler Universität, A-4040 Linz, AustriaView Profile Authors Info & Claims ISSAC '91: Proceedings of the 1991 international symposium on Symbolic and algebraic computationJune 1991 Pages 403–410https://doi.org/10.1145/120694.120756Online:01 June 1991Publication History 9citation154DownloadsMetricsTotal Citations9Total Downloads154Last 12 Months2Last 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

The Knuth-Bendix procedure for the completion of a rewrite rule system and the Buchberger algorithm for computing a Gröbner basis of a polynomial ideal are very similar in two respects: they both start with an arbitrary specification of an algebraic structure (axioms for an equational theory and a basis for a polynomial ideal, respectively) which is transformed to a very special specification of this algebraic structure (a complete rewrite rule system and a Gröbner basis of the polynomial ideal, respectively). This special specification allows to decide many problems concerning the given algebraic structure. Moreover, both algorithms achieve their goals by employing the same basic concepts: formation of critical pairs and completion.

article Free AccessArtifacts AvailableArtifacts Evaluated & Reusable Share on Algorithm 628: An algorithm for constructing canonical bases of polynomial ideals Authors: F. Winkler Univ. of Delaware, Newark Univ. of Delaware, NewarkView Profile , B. Buchberger Johannes Kepler Univ., Linz, Austria Johannes Kepler Univ., Linz, AustriaView Profile , F. Lichtenberger Johannes Kepler Univ., Linz, Austria Johannes Kepler Univ., Linz, AustriaView Profile , H. Rolletschek Kent State Univ., Kent, OH Kent State Univ., Kent, OHView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 11Issue 1March 1985 pp 66–78https://doi.org/10.1145/3147.214316Online:01 March 1985Publication History 14citation472DownloadsMetricsTotal Citations14Total Downloads472Last 12 Months13Last 6 weeks1 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 Alerts New Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF