Papers
The Dixon Resultant
expository paper
This is an expository paper explaining the Dixon resultant, as extended by Kapur, Saxena, and Yang.
In most circumstances, this is the method of choice in solving systems of multivariate polynomials. I would except from that statement very large systems that come up in cryptography.
Fermat code to run Dixon can be found here:
http://home.bway.net/lewis/dixon
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Apollonius Problems in Biochemistry
coauthor S. Bridgett. published in MATCOM 61(2) p. 101 - 114, 2003
The Apollonius Circle Problem dates to Greek antiquity, circa 250 BC. Given three circles in the plane, find or construct a circle tangent to all three. This was generalized by replacing some circles with straight lines. Descartes (and many later people) considered a special case in which all four circles are mutually tangent to each other (i.e. pairwise). In this paper we consider the general case in two and three dimensions, and further generalizations with ellipsoids and lines. We believe we are the first to explicitly find the polynomial equations for the parameters of the solution sphere in these generalized cases. Doing so is quite a challenge for the best computer algebra systems. We report below some comparative times using various computer algebra systems on some of these problems. We also consider conic tangency equations for general conics in two and three dimensions.
Apollonius problems are of interest in their own right. However,
the motivation for this work came originally from medical research, specifically the problem of computing the medial axis of the space around a molecule: obtaining the position and radius of a sphere which touches four known spheres or ellipsoids.
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Algorithmic Search for Flexibility using Resultants of Polynomial Systems
coauthor E. Coutsias. published in Automated Deduction in Geometry. Lecture Notes in Computer Science, Vol. 4869, p. 68 - 79. Springer
This paper describes the recent convergence of four topics: polynomial systems, flexibility of three dimensional objects, computational chemistry, and computer algebra. We discuss a way to solve systems of polynomial equations with resultants. Using ideas of Bricard, we find a system of polynomial equations that models a configuration of quadrilaterals that is equivalent to some three dimensional structures. These structures are of interest in computational chemistry, as they represent molecules. We then describe an algorithm that examines the resultant and determines ways that the structure can be flexible.
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Heuristics to Accelerate the Dixon Resultant
published in MATCOM 77, Issue 4, April 2008
The Dixon Resultant method solves a system of polynomial equations by computing its resultant. It constructs a square matrix whose determinant det is a multiple of the resultant res. The naive way to proceed is to compute det, factor it, and identify res. But often det is too large to compute or factor, even though res is relatively small.
In this paper we describe three heuristic methods that often overcome these problems. The first, although sometimes useful by itself, is often a subprocedure of the second two. The second may be used on any polynomial system to discover factors of det without producing the complete determinant. The third applies when res appears as a factor of det in a certain exponential pattern. This occurs in some symmetrical systems of equations. We show examples from computational chemistry, signal processing, dynamical systems, quantifier elimination, and pure mathematics.
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Solving the Recognition Problem for Six Lines Using the Dixon Resultant
coauthor Peter Stiller. MATCOM 49 (1999)
The “Six-Line Problem” arises in computer vision and in the automated analysis of images. Given a three-dimensional object, one extracts geometric features (for example six lines) and then, via techniques from algebraic geometry and geometric invariant theory, produces a set of three-dimensional invariants that represents that feature set. Suppose that later an object is encountered in an image. (For example a photograph taken by a camera modeled by standard perspective projection, i.e. a “pinhole” camera.) Suppose further that six lines are extracted from the object appearing in the image.The problem is to decide if the object in the image is the original 3D object.To answer this question two-dimensional invariants are computed from the lines in the image.One can show that conditions for geometric consistency between the three-dimensional object features and the two dimensional image features can be expressed as a set of polynomial equations in the combined set of two and three dimensional invariants.The object in the image is geometrically consistent with the original object if the set of equations has a solution.One well known method to attack such sets of equations is with resultants. Unfortunately, the size and complexity of this problem made it appear overwhelming until recently. This paper will describe a solution obtained using our own variant of the Cayley-Dixon-Kapur-Saxena-Yang resultant. There is reason to suspect that the resultant technique we employ here may solve other complex polynomial systems.
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Solving the Least Squares Method Problem in the AHP for 3 x 3 and 4 x 4 Matrices
coauthor S. Bozoki. published in Central European Journal for Operations Research, September 2005
The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM) are of the possible tools for computing the priorities of the alternatives. A method for generating all the solutions of the LSM problem for 3 × 3 and 4 × 4 matrices is discussed in the paper. Our algorithms are based on the theory of resultants.
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Computer Search for Nilpotent Complexes
published in Experimental Mathematics, 1997
The concept of nilpotency for a topological space is a generalization of simple connectivity. That it is a fruitful generalization was shown by Dror, Kan, Bousfield, Hilton, and others. In 1977 Brown and Kahn proved that the dimension of a nilpotent complex can be read from the ordinary homology groups, just as in the case of a simply connected complex. They also showed that if a nilpotent complex has finite and nontrivial fundamental group, its dimension must be at least 3. In 1985 Lewis showed that for any finite nilpotent group there is a (not necessarily finite) three-dimensional nilpotent complex with that fundamental group. The smallest finite nilpotent group for which it was unknown whether a finite three dimensional nilpotent complex exists was Z2+Z6. The authors, together with a team of undergraduate students at Fordham University, used computers to search for three dimensional finite nilpotent complexes over groups of the form Zn+Zm. Such complexes were eventually found for Z2+Z6, Z2+Z10, and Z3+Z6.
This article describes the strategy for constructing nilpotent complexes of dimension three, and some of the issues in implementing the computer search. The main computational issues are “normalizing” matrices, especially to the Smith normal form, and mapping matrices over Z to matrices over Zp for various primes p. We conclude with a summary of the complexes discovered and open questions.
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Isomorphism Classes and Derived Series of Certain Almost-Free Groups
published in Experimental Mathematics, 1994
Baumslag defined a family of groups that are of interest because they closely resemble free groups, yet are not free. It was known that each group in this family has the same lower central series of quotients and the same first two terms in the derived series of quotients as does the free group F on two generators. We have verified that there are different isomorphism types among the groups in the family, and that the third terms in the derived series of quotients are often distinct from that of F. Our basic technique is to count the number of homomorphisms from the groups of interest to a target group.
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