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Algorithmic Elimination Methods Algorithmic Elimination Methods Deepak Kapur∗ Institute for Programming and Logics Department of Computer Science State University of New York Albany, NY 12222 [email protected] Very rough draft: June 19, 1995y Abstract This tutorial provides a gentle introduction to different methods for eliminating variables from a finite system of polynomials and for solving polynomial equations. Elimination methods have applications in many fields. The first part of the tutorial will discuss resultant-based methods investigated in the 19th century and early 20th century. The main idea is to generate from a system of nonlinear polynomial equations, possibly a larger system of independent polynomial equations. The larger system has as many equations as distinct terms in the polyno- mials. Each term is treated as an unknown and the theory of linear algebra can be applied. Three major resultant methods due to Euler, Bezout, Sylvester, Cayley, Dixon and Macaulay, and a sparse reformulation of Macaulay will be discussed and compared. In the second part, two other methods { Gr¨obnerbasis method based on polyno- mial ideal theory and the characteristic set method based on algebraic geometry will be reviewed. The use of these methods for elimination and equation solving will be discussed. Examples from application domains such as computer vision, geometric reasoning and solid modeling will be used. These notes discuss resultants and characteristic set methods as written material on these topics is not easily available. Discussion on Gr¨obnerbasis methods is omit- ted because there are now good books introducing Gr¨obnerbasis methods as well as discussing Gr¨obnerbasis theory in depth. ∗Supported in part by a grant from United States Air Force Office of Scientific Research AFOSR-91-0361. yThese notes are expected to be revised extensively based on feedback and comments from the participants in the ISSAC tutorial in Montreal. The revised version would be made available by anonymous ftp. Until then, the distribution of these notes is restricted to the tutorial participants. Please get the author's permission before making additional copies and distributing it your colleagues. 1 1 Introduction Nonlinear polynomials are used to model many physical phenomena in engineering appli- cations. Often there is a need to solve nonlinear polynomial equations, or if solutions do not have to be explicitly computed, it is necessary to study the nature of solutions. Many engineering problems can be easily formulated using polynomials with the help of extra variables. It then becomes necessary to eliminate some of those extra variables. Examples include implicitization of curves and surfaces, invariants with respect to transformation groups, robotics and kinematics. Projection of a variety (which is the set of solutions of polynomial equations in an algebraically closed field) is yet another useful tool in algebraic geometry. These applications require the use of elimination techniques on polynomials, and symbolically studying the solutions of the corresponding equations. To get an idea about the use of polynomials for modeling in different application domains, the reader may consult books by Hoffman [14], Abhyankar [2], Morgan [24], Kapur and Mundy [17], Donald, Kapur and Mundy [11]. In these notes, we discuss different symbolic methods for studying the solutions of poly- nomial equations. Because of lack of time and space, we are unable to discuss numerical techniques for solving polynomial equations. An interested reader may want to look at [28] for a simple approach based on interval methods and branch and prune techniques, that has been found to be quite effective. The method is compared with homotopy based continuation methods [24] as well as other interval based techniques. In the first part of these notes, we focus on resultant methods. Resultant methods were developed in the 18th century by Newton, Euler and Bezout, in the 19th century by Sylvester and Cayley, and the early parts of 20th century by Dixon and Macaulay. These constructivists were attempting to extend simple linear algebra techniques developed for linear equations to nonlinear polynomial equations. A number of books on the theory of equations were written which are now out of print. Some very interesting subsections were omitted by the authors in revised editions of some of these books because abstract methods in algebraic geometry and invariant theory gained dominance, and constructive methods began to be viewed as too concrete to provide any useful structural insights. A good example is the 1970 edition of van der Waerden's book on algebra which does not include a beautiful chapter on elimination theory that appeared as the first chapter in the second volume of its 1940 edition. For an excellent discussion of the history of constructive methods in algebra, the reader is referred to an article by Professor Abhyankar [1]. In the second part of this paper, we discuss Ritt's characteristic set construction and its popularization by Wu Wen-tsun. Despite its tremendous success in geometry theorem proving, characteristic set methods do not seem to have gotten as much exposure as they should. For many problems, characteristic set construction is quite efficient and powerful. In these notes, we have focussed on the problems of computing zeros of polynomials, elimination of variables and projection of the the zeros of polynomials along certain dimen- sions. We have attempted to provide information that is not easily available; otherwise, we have assumed that an interested reader can consult various references for further details. For instance, we have chosen not to include a section on Gr¨obnerbasis constructions and the related theory. For an introductory reading, an interested reader may consult Buchberger's survey articles [4, 5], a section in our introductory paper on elimination methods [16], 2 Hoffman’s book [14], or Cox et al's book [9]. For a comprehensive treatment, the reader can consult Weispfenning's book [29]. For a discussion of characteristic set construction, the reader may consult Wu's articles [31, 32], Chou's book [8] or a section in our introductory article [16]. In these notes, we have added some new material on characteristic sets which could not be included in the introductory article because of lack of space. Most of the material in these notes is on resultants since except for Sylvester's resultants, very little material is included in text books. In order to conserve space, we have been selective in including examples to illustrate different concepts. In the oral presentation at the tutorial, we plan to use many examples { simple as well as nontrivial ones from different application domains. Examples have not been included to illustrate Sylvester resultants for the univariate case, Dixon's resultants for the bivariate case, or Macaulay's multivariate resultants as these examples are given in detail in our introductory paper [16]; an interested reader can consult [16] for examples on these topics. We have, however, included examples to illustrate sparse resultants as well as the new material on Dixon resultants. 1.1 Overview In the next subsection, we provide basic background information about zeros of polynomials, distinction between affine zeros and projective zeros, and the zero set of a set of polynomials. The next section is about resultants. We first review Euler's method for computing the resultant of two polynomials by eliminating one variable, Sylvester's reformulation using di- alytic expansion, Macaulay's generalization for simultaneously eliminating many variables. U-resultants are reviewed to discuss how common zeros can be extracted using resultants. The second part starts with Bezout's formulation of the resultant of two polynomials by eliminating one variable. This is followed by Cayley's remarkable formulation of Bezout's method which naturally generalizes nicely to a formulation for simultaneously eliminating many variables. Dixon's formulation of the resultant for three polynomials in two vari- ables is first discussed. Then this formulation is generalized. Two methods - generalized characteristic polynomial method based on perturbation and the rank submatrix method, are reviewed for extracting a projection operator from singular resultant matrices. An experimental comparison of different resultant formulations is given in the form of a table presenting performance of different method on examples from different application domains. Some recent results on theoretical complexity of Dixon resultants are included. The reader would note that the formulations in this section are expressed in terms of appropriate determinants which correspond to solving a linear system of equations in which the unknowns are the power products appearing in the polynomials. This linear system is obtained by transforming the original set of polynomial equations. This is in contrast to Gr¨obnerbasis and characteristic methods, where no system of linear equations is explicitly and a priori set up. Instead, these methods appear to be similar to completion algorithms in which new polynomials are successively constructed until a subset of polynomials with the desired property is generated. The third section discusses Ritt's characteristic sets. We have pointed fundamental differences in Ritt's definition and the definition used by Wu in his work. We discuss the use of characteristic set method for elimination. As an alternative to decomposition of a zero set into irreducible zero sets based on Rit-Wu's algorithm, a different method using Duval's D5 algorithm is presented for decomposing a zero set. 3 1.2 Preliminaries Let Q , IR, C, respectively, denote the fields of rational numbers, real numbers, and complex numbers. Unless specified otherwise, by a polynomial, we mean a multivariate polynomial with rational coefficients. A multivariate polynomial can be viewed as a sum of products with nonzero rational coefficients, where each product is called a term or power product; together with its coefficient, it is called a monomial. A univariate polynomial p(x) is said to vanish at a value a if the polynomial p evaluates to zero when a is uniformly substituted for x in p.
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