Algorithms for eliminating variables from systems of multivariate polynomials are essential tools in constructive algebra and algebraic geometry. The reason is that a number of important computational problems in these areas can be tackled by elimination techniques. In particular, elimination methods can be used for solving systems of polynomial equations, a classical mathematical problem with a wide range of applications in many areas of science, engineering and industry. It is therefore not surprising that elimination theory has always played a central role in computer algebra and that techniques which resulted from continuing research efforts on this subject are the basis of a number of practical algorithms in current computer algebra systems.
To review the state-of-the-art, report recent developments and exchange views on future research, a special session on the subject was organized by the second editor at the 1996 IMACS conference on Applications of Computer Algebra. The presentations at the conference showed that there are many significant new results including novel ideas and advances on algorithms and software tools for polynomial elimination and the applications thereof. The present special issue has been prepared to collect and publish such results. The interest and productivity of research on polynomial elimination is reflected, in part, by the number of talks given at the IMACS conference and the number of submissions to this special issue: the former is 11 and the latter is 27. After a strict refereeing and revision process, according to the usual standard of the Journal of Symbolic Computation, β 13 papers have been accepted for publication in this special issue. These papers represent some of the trends and achievements of research on the subject in the past few years. We hope that this special issue will serve as a reference source for further investigations.
The first paper is a survey by Emiris and Mourrain that unifies the existing work on resultants, with emphasis on matrix constructions which generalize the classical concepts. The authors present and compare the different formulations of matrices named after Sylvester, BΓ©zout, Macaulay and Dixon, as well as toric resultant matrices, and explain how these matrices can be used for solving concrete problems.
The aim of the paper by Bikker and Uteshev is to interpret some of the original ideas of BΓ©zout and his successors in the language of modern mathematics and to explore links to recent investigations in elimination theory. In particular, the problem of finding an eliminant for a system of multivariate equations and representations for its zeroes as rational functions of the roots of that eliminant is discussed. Gonzalez-Vega and Gonzalez-Campos introduce one procedure for eliminating single variables for poly- β The two papers authored by the Editors were handled similarly to all other submitted manuscripts. They were refereed by anonymous referees, whose identities are unknown to the authors, and a minimum of two referees recommended publication.