Algebraic and geometric reasoning using Dixon resultants

Kapur, Deepak; Saxena, Tripti; Yang, Lu

doi:10.1145/190347.190372

Cited by 105 publications

(74 citation statements)

References 7 publications

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“…This result, although a direct consequence of Proposition 3.11, is new, to the best of our knowledge. It generalizes the result of Kapur et al (1994) where, under a technical hypothesis, a multiple of the toric resultant is computed. is divisible by the resultant Res X (f 1 , .…”

Section: Proposition 312 Any Non-zero Minor Of Maximal Size Of the supporting

confidence: 56%

“…As mentioned at the beginning, this tool is fundamental in many recent algorithmic developments (Aizenberg and Kytmanov, 1981;Berenstein and Yger, 1991;Berenstein et al, 1993;Cardinal, 1993;Fitchas et al, 1993;Kapur et al, 1994Kapur et al, , 1996Becker et al, 1996;Cardinal and Mourrain, 1996;Kapur and Saxena, 1997;Mourrain, 1998, 1999).…”

Section: Bézout Matricesmentioning

confidence: 99%

“…These numerical matrix manipulations manage to obtain smaller matrices that contain all relevant information through a Schur factorization, Jordan decomposition or singular value decomposition, respectively (Manocha and Demmel, 1995;Corless et al, 1997;Mourrain, 1998), or computing maximal non-singular minors (Kapur et al, 1994;Cardinal and Mourrain, 1996).…”

Section: System Solving By Eigenvectorsmentioning

confidence: 99%

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Matrices in Elimination Theory

Emiris

Mourrain

1999

Journal of Symbolic Computation

104

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The last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in a number of scientific and engineering applications. On the other hand, the Bezoutian reveals itself as an important tool in many areas connected to elimination theory and has its own merits, leading to new developments in effective algebraic geometry. This survey unifies the existing work on resultants, with emphasis on constructing matrices that generalize the classic matrices named after Sylvester, Bézout and Macaulay. The properties of the different matrix formulations are presented, including some complexity issues, with emphasis on variable elimination theory. We compare toric resultant matrices to Macaulay's matrix and further conjecture the generalization of Macaulay's exact rational expression for the resultant polynomial to the toric case. A new theorem proves that the maximal minor of a Bézout matrix is a non-trivial multiple of the resultant. We discuss applications to constructing monomial bases of quotient rings and multiplication maps, as well as to system solving by linear algebra operations. Lastly, degeneracy issues, a major preoccupation in practice, are examined. Throughout the presentation, examples are used for illustration and open questions are stated in order to point the way to further research.

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Section: Proposition 312 Any Non-zero Minor Of Maximal Size Of the supporting

confidence: 56%

Section: Bézout Matricesmentioning

confidence: 99%

Section: System Solving By Eigenvectorsmentioning

confidence: 99%

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Matrices in Elimination Theory

Emiris

Mourrain

1999

Journal of Symbolic Computation

104

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“…• However, we may continue [12], [4]: Find any maximal rank submatrix; let ksy = its determinant. Existence of a common solution implies ksy = 0.…”

Section: The Cayley-dixon-bezout-ksy Resultant Methodsmentioning

confidence: 99%

“…If this Gröbner Basis is chosen carefully, it will contain the desired resultant. Resultant methods are less well known, especially the apparent method of choice, the Bezout-Cayley-Dixon-KSY method [12], [14], [4], which we describe below.…”

Section: To Accomplish This Elimination We Use Eithermentioning

confidence: 99%

Conic tangency equations and Apollonius problems in biochemistry and pharmacology

Lewis

Bridgett²

2003

Mathematics and Computers in Simulation

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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. Vieta [19] (or Viète) solved the problem using circle inversions before 1580. Two generations later, Descartes 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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Resultants and loop closure

Coutsias

Seok

Wester

et al. 2005

Int J of Quantum Chemistry

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ABSTRACT:The problem of tripeptide loop closure is formulated in terms of the angles { i } iϭ1 3 describing the orientation of each peptide unit about the virtual axis joining the C ␣ atoms. Imposing the constraint that at the junction of two such units the bond angle between the bonds C ␣ ON and C ␣ OC is fixed at some prescribed value results in a system of three bivariate polynomials in u i :ϭ tan i /2 of degree 2 in each variable. The system is analyzed for the existence of common solutions by making use of resultants, determinants of matrices composed of the coefficients of two (or more) polynomials, whose vanishing is a necessary and sufficient condition for the polynomials to have a common root. Two resultants are compared: the classical Sylvester resultant and the Dixon resultant. It is shown that when two of the variables are eliminated in favor of the third, a polynomial of degree 16 results. To each one of its real roots, there is a corresponding common zero of the system. To each such zero, there corresponds a consistent conformation of the chain. The Sylvester method can find these zeros among the eigenvalues of a 24 ϫ 24 matrix. For the Dixon approach, after removing extraneous factors, an optimally sized eigenvalue problem of size 16 ϫ 16 results. Finally, the easy extension to the more general problem of triaxial loop closure is presented and an algorithm for implementing the method on arbitrary chains is given.

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Algebraic and geometric reasoning using Dixon resultants

Cited by 105 publications

References 7 publications

Matrices in Elimination Theory

Matrices in Elimination Theory

Conic tangency equations and Apollonius problems in biochemistry and pharmacology

Resultants and loop closure

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