Search

A d-dimensional tensor A of format n × n × ⋯ × n defines naturally a rational map Ψ from the projective space P n − 1 to itself and its eigenscheme is then the subscheme of P n − 1 of fixed points of Ψ. The eigendiscriminant is an irreducible polynomial in the coefficients of A that vanishes for a given tensor if and only if its eigenscheme is singular. In this paper, we contribute two formulas for the computation of eigendiscriminants in the cases n = 3 and n = 4. In particular, by restriction to symmetric tensors, we obtain closed formulas for the eigendiscriminants of plane curves and surfaces in P 3 as the ratio of some determinants of resultant matrices. [ABSTRACT FROM AUTHOR]

A triangle center such as the incenter, barycenter, etc., is specified by a function thrice- and cyclically applied on sidelengths and/or angles. Consider the 1d family of 3-periodics in the elliptic billiard, and the loci of its triangle centers. Some will sweep ellipses, and others higher-degree algebraic curves. We propose two rigorous methods to prove if the locus of a given center is an ellipse: one based on computer algebra, and another based on an algebro-geometric method. We also prove that if the triangle center function is rational on sidelengths, the locus is algebraic. [ABSTRACT FROM AUTHOR]

The square-root V'elu's formula (√'elu's formula) is known to be effective way to speed up the computations of higher-degree isogeny used for isogeny-based cryptosystems such as CSIDH and B-SIDH. The original formula was proposed using Montgomery curves, and Moriya et al. then extended it to Edwards curves without specific construction method. In this study, we explicitly show how to construct√'elu's formula on Edwards curves. In particular, we provide a method for adjusting for the differences between two resultants. We also compare the computational complexity with the original√'elu's formula and find that it is up to 8% faster. [ABSTRACT FROM AUTHOR]

We give combinatorial generalizations of the Cayley--Bacharach theorem and induced map. [ABSTRACT FROM AUTHOR]

We compute and study two determinantal representations of the discriminant of a cubic quaternary form. The first representation is the Chow form of the 2-uple embedding of ℙ³ and is computed as the Pfaffian of the Chow form of a rank 2 Ulrich bundle on this Veronese variety. We then consider the determinantal representation described by Nanson. We investigate the geometric nature of cubic surfaces whose discriminant matrices satisfy certain rank conditions. As a special case of interest, we use certain minors of this matrix to suggest equations vanishing on the locus of k-nodal cubic surfaces. [ABSTRACT FROM AUTHOR]

We formalize algebraic numbers in Isabelle/HOL. Our development serves as a verified implementation of algebraic operations on real and complex numbers. We moreover provide algorithms that can identify all the real or complex roots of rational polynomials, and two implementations to display algebraic numbers, an approximative version and an injective precise one. We obtain verified Haskell code for these operations via Isabelle's code generator. The development combines various existing formalizations such as matrices, Sturm's theorem, and polynomial factorization, and it includes new formalizations about bivariate polynomials, unique factorization domains, resultants and subresultants. [ABSTRACT FROM AUTHOR]

Hyperbolic polynomials are real multivariate polynomials with only real roots along a fixed pencil of lines. Testing whether a given polynomial is hyperbolic is a difficult task in general. We examine different ways of translating hyperbolicity into nonnegativity conditions, which can then be tested via sum-of-squares relaxations. [ABSTRACT FROM AUTHOR]

The explicit symplectic difference schemes are considered for the numerical solution of molecular dynamics problems described by systems with separable Hamiltonians. A general method for finding symplectic schemes of high order of accuracy using parametric Gröbner bases, resultants, permutations of variables, and optimization techniques is proposed. The implementation of the method is described by the example of four-stage partitioned Runge–Kutta (PRK) schemes of the Forest–Ruth and Yoshida families. 97 new PRK schemes of Forest–Ruth family have been obtained. The error functionals for five new schemes FR47, FR50, FR51, FR52, and FR59 are smaller by more than two orders of magnitude than in the case of the fourth-order schemes, which were obtained previously in the explicit form in this family. In the family of Yoshida schemes, 115 new schemes have been found. Verification of the schemes was carried out on the Kepler's problem, which has the exact solution. Calculations of this task performed at large time steps showed that the FR50 scheme significantly exceeds in terms of accuracy the four-stage Forest–Ruth schemes known in the literature, the Verlet scheme, schemes from the Yoshida family Y7, Y95, Y110, and a six-stage fourth-order scheme. • The use of the parametric Gröbner bases and resultants is shown to facilitate the obtaining of new symplectic schemes. • An efficient two-step optimization algorithm is proposed for obtaining new symplectic schemes. • 97 new four-stage fourth-order symplectic schemes were found in the Forest-Ruth family of partitioned Runge-Kutta schemes. • 115 new four-stage fourth-order symplectic schemes were found in the Yoshida family of partitioned Runge-Kutta schemes. [ABSTRACT FROM AUTHOR]

The problem of tripeptide loop closure is formulated in terms of the angles {Ti}(i)(3)(=1), describing the orientation of each peptide unit about the virtual axis joining the C alpha atoms. Imposing the constraint that at the junction of two such units the bond angle between the bonds C alpha-N and C alpha-C is fixed at some prescribed value 0 results in a system of three bivariate polynomials in u(i) : = tan tau(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 X 24 matrix. For the Dixon approach, after removing extraneous factors, an optimally sized eigenvalue problem of size 16 X 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. (c) 2005 Wiley Periodicals, Inc.

Let A be a commutative ring with identity and be an algebraic element over A. We give necessary and sufficient conditions under which the simple algebraic extension A[α] is without nilpotent or without idempotent elements.

We provide irreducibility criteria for multiplicative convolutions of polynomials with integer coefficients, that is, for polynomials of the form hdeg f · f(g/h), where f, g, h are polynomials with integer coefficients, and g and h are relatively prime. The irreducibility conditions are expressed in terms of the prime factorization of the leading coefficient of the polynomial hdeg f · f(g/h), the degrees of f, g, h, and the absolute values of their coefficients. In particular, by letting h = 1 we obtain irreducibility conditions for compositions of polynomials with integer coefficients.

Given n ∈ N , we study the conditions under which a finite field of prime order q will have adjacent elements of multiplicative order n . In particular, we analyze the resultant of the cyclotomic polynomial Φ n ( x ) with Φ n ( x + 1 ) , and exhibit Lucas and Mersenne divisors of this quantity. For each n ≠ 1 , 2 , 3 , 6 , we prove the existence of a prime q n for which there is an element α ∈ Z q n where α and α + 1 both have multiplicative order n . Additionally, we use algebraic norms to set analytic upper bounds on the size and quantity of these primes. [ABSTRACT FROM AUTHOR]

We construct a certain $${\mathbb{F}_{2}}$$ -valued analogue of the mixed volume of lattice polytopes. This 2-mixed volume cannot be defined as a polarization of any kind of an additive measure, or characterized by any kind of its monotonicity properties, because neither of the two makes sense over $${\mathbb{F}_2}$$ . In this sense, the convex-geometric nature of the 2-mixed volume remains unclear. On the other hand, the 2-mixed volume seems to be no less natural and useful than the classical mixed volume-in particular, it also plays an important role in algebraic geometry. As an illustration of this role, we obtain a closed-form expression in terms of the 2-mixed volume to compute the signs of the leading coefficients of the resultant, which were by now explicitly computed only for some special cases. [ABSTRACT FROM AUTHOR]

It is well known that resultant elimination is an effective method of solving multivariate polynomial equations. In this paper, instead of computing the target resultants via variable by variable elimination, the authors combine multivariate implicit equation interpolation and multivariate resultant elimination to compute the reduced resultants, in which the technique of multivariate implicit equation interpolation is achieved by some high probability algorithms on multivariate polynomial interpolation and univariate rational function interpolation. As an application of resultant elimination, the authors illustrate the proposed algorithm on three well-known unsolved combinatorial geometric optimization problems. The experiments show that the proposed approach of resultant elimination is more efficient than some existing resultant elimination methods on these difficult problems. [ABSTRACT FROM AUTHOR]

A resultant-based method to calculate the overdetermined strata for degree 6 hyperbolic polynomials in one variable is revealed. This method is a new method to calculate overdetermined strata. The overdetermined strata in degree 6 have not been calculated before as the geometric method used until now can not be generalized to degree n ≥ 6. [ABSTRACT FROM AUTHOR]

Hidden-variable resultant methods are a class of algorithms for solving multidimensional polynomial rootfinding problems. In two dimensions, when significant care is taken, they are competitive practical rootfinders. However, in higher dimensions they are known to miss zeros, calculate roots to low precision, and introduce spurious solutions. We show that the hidden variable resultant method based on the Cayley (Dixon or Bézout) matrix is inherently and spectacularly numerically unstable by a factor that grows exponentially with the dimension. We also show that the Sylvester matrix for solving bivariate polynomial systems can square the condition number of the problem. In other words, two popular hidden variable resultant methods are numerically unstable, and this mathematically explains the difficulties that are frequently reported by practitioners. Regardless of how the constructed polynomial eigenvalue problem is solved, severe numerical difficulties will be present. Along the way, we prove that the Cayley resultant is a generalization of Cramer's rule for solving linear systems and generalize Clenshaw's algorithm to an evaluation scheme for polynomials expressed in a degree-graded polynomial basis. [ABSTRACT FROM AUTHOR]

We develop tools from computational algebraic geometry for the study of steady state features of autonomous polynomial dynamical systems via elimination of variables. In particular, we obtain nontrivial bounds for the steady state concentration of a given species in biochemical reaction networks with mass-action kinetics. This species is understood as the output of the network and we thus bound the maximal response of the system. The improved bounds give smaller starting boxes to launch numerical methods. We apply our results to the sequential enzymatic network studied in Markevich et al. (J Cell Biol 164(3):353-359, ) to find nontrivial upper bounds for the different substrate concentrations at steady state. Our approach does not require any simulation, analytical expression to describe the output in terms of the input, or the absence of multistationarity. Instead, we show how to extract information from effectively computable implicit dose-response curves, with the use of resultants and discriminants. We moreover illustrate in the application to an enzymatic network, the relation between the exact implicit dose-response curve we obtain symbolically and the standard hysteresis diagram provided by a numerical ode solver. The setting and tools we propose could yield many other results adapted to any autonomous polynomial dynamical system, beyond those where it is possible to get explicit expressions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and compare three different methods of computing the intersections of rational parametrized plane curves. The common approach of these methods is to apply the μ-basis of the plane curves, and avoid of computing the implicit equations of the curves, which increase the computation efficiency. [ABSTRACT FROM AUTHOR]

Abstract: In this article we report on our experience in computing resultants of bivariate polynomials on Graphics Processing Units (GPU). Following the outline of Collins’ modular approach [6], our algorithm starts by mapping the input polynomials to a finite field for sufficiently many primes . Next, the GPU algorithm evaluates the polynomials at a number of fixed points , and computes a set of univariate resultants for each modular image. Afterwards, the resultant is reconstructed using polynomial interpolation and Chinese remaindering. The GPU returns resultant coefficients in the form of Mixed Radix (MR) digits. Finally, large integer coefficients are recovered from the MR representation on the CPU. All computations performed by the algorithm (except for, partly, Chinese remaindering) are outsourced to the graphics processor thereby minimizing the amount of work to be done on the host machine. The main theoretical contribution of this work is the modification of Collins’ modular algorithm using the methods of matrix algebra to make an efficient realization on the GPU feasible. According to the benchmarks, our algorithm outperforms a CPU-based resultant algorithm from 64-bit Maple 14 by a factor of 100. [Copyright &y& Elsevier]

Abstract: A trigonometric polynomial is a truncated Fourier series of the form . It has been previously shown by the author that zeros of such a polynomial can be computed as the eigenvalues of a companion matrix with elements which are complex valued combinations of the Fourier coefficients, the “CCM” method. However, previous work provided no examples, so one goal of this new work is to experimentally test the CCM method. A second goal is introduce a new alternative, the elimination/Chebyshev algorithm, and experimentally compare it with the CCM scheme. The elimination/Chebyshev matrix (ECM) algorithm yields a companion matrix with real-valued elements, albeit at the price of usefulness only for real roots. The new elimination scheme first converts the trigonometric rootfinding problem to a pair of polynomial equations in the variables where and . The elimination method next reduces the system to a single univariate polynomial . We show that this same polynomial is the resultant of the system and is also a generator of the Groebner basis with lexicographic ordering for the system. Both methods give very high numerical accuracy for real-valued roots, typically at least 11 decimal places in Matlab/IEEE 754 16 digit floating point arithmetic. The CCM algorithm is typically one or two decimal places more accurate, though these differences disappear if the roots are “Newton-polished” by a single Newton’s iteration. The complex-valued matrix is accurate for complex-valued roots, too, though accuracy decreases with the magnitude of the imaginary part of the root. The cost of both methods scales as floating point operations. In spite of intimate connections of the elimination/Chebyshev scheme to two well-established technologies for solving systems of equations, resultants and Groebner bases, and the advantages of using only real-valued arithmetic to obtain a companion matrix with real-valued elements, the ECM algorithm is noticeably inferior to the complex-valued companion matrix in simplicity, ease of programming, and accuracy. [Copyright &y& Elsevier]

We study combinatorial and number-theoretic issues arising from group-invariant CR mappings from spheres to hyperquadrics. Given an initial complex analytic function, we define two sequences of complex numbers. We derive formulae for these sequences in terms of the Taylor coefficients of the initial function and, in the polynomial case, also in terms of the reciprocals of the roots. We connect these formulae to CR mappings invariant under arbitrary representations of finite cyclic subgroups of the unitary group. We prove that the resultant of them-th polynomial (which in all cases has integer coefficients) and any of its partial derivatives is divisible bymm. [ABSTRACT FROM AUTHOR]

Abstract: We present a new and complete algorithm for computing the topology of an algebraic surface given by a square free polynomial in . Our algorithm involves only subresultant computations and entirely relies on rational manipulation, which makes it direct to implement. We extend the work in , on the topology of non-reduced algebraic space curves, and apply it to the polar curve or apparent contour of the surface . We exploit a simple algebraic criterion to certify the pseudo-genericity and genericity position of the surface. This gives us rational parametrizations of the components of the polar curve, which are used to lift the topology of the projection of the polar curve. We deduce the connection of the two-dimensional components above the cell defined by the projection of the polar curve. A complexity analysis of the algorithm is provided leading to a bound in for the complexity of the computation of the topology of an implicit algebraic surface defined by integer coefficient polynomial of degree and coefficient size . Examples illustrate the implementation in Mathemagix of this first complete code for certified topology of algebraic surfaces. [Copyright &y& Elsevier]

Abstract: For a semialgebraic set in , let be the cone of polynomials in of degrees at most that are nonnegative on . This paper studies the geometry of its boundary . We show that when and is even, its boundary lies on the irreducible hypersurface defined by the discriminant of . We show that when is a real algebraic variety, lies on the hypersurface defined by the discriminant of . We show that when is a general semialgebraic set, lies on a union of hypersurfaces defined by the discriminantal equations. Explicit formulae for the degrees of these hypersurfaces and discriminants are given. We also prove that typically does not have a barrier of type when is required to be a polynomial, but such a barrier exists if is allowed to be semialgebraic. Some illustrating examples are shown. [Copyright &y& Elsevier]

While the recent explosion of affordable software has placed powerful electronic music production tools within reach of anyone with a smart phone or a tablet, there are important aspects of music composition that technology has failed to address. In fact, much audio software subtly encourages prefabricated aesthetic solutions via the user interface, resulting in a cookie-cutter approach to music creation. As a result, much of the electronic music we hear today rarely ventures into new territory by exploring new structures. One solution could be to employ a compositional tool that enables a composer to generate new and unusual structures derived from mathematical relationships such as the Fibonacci series, the Golden Ratio or other mathematical patterns found in nature. Luckily, such a tool exists and is part of a vast system of music composition created in the twentieth century by the visionary music theoretician and composer Joseph Schillinger. In this article the author explores a Schillinger technique called ‘composing with density groups’ which can be used to create unique musical structures on paper or in any digital audio workstation (DAW). [ABSTRACT FROM PUBLISHER]

Abstract: Sylvester double sums, introduced first by Sylvester (see ), are symmetric expressions of the roots of two polynomials, while subresultants are defined through the coefficients of these polynomials (see and for references on subresultants). As pointed out by Sylvester, the two notions are very closely related: Sylvester double sums and subresultants are equal up to a multiplicative non-zero constant in the ground field. Two proofs are already known: that of , using Schur functions, and that of , using manipulations of matrices. The purpose of this paper is to give a new simple proof using similar inductive properties of double sums and subresultants. [ABSTRACT FROM AUTHOR]

Abstract: We revisit the Kukles system to show how advances in computer hardware and software have improved symbolic calculations. We confirm directly that there are no non-persistent centres for the Kukles system. We also prove an earlier conjecture regarding the isochronous centres for the extended Kukles system. We introduce a technique whereby modular resultants can be used to investigate the properties of resultants that cannot be readily calculated using the existing software. [ABSTRACT FROM AUTHOR]

We give a simple inductive proof of quadratic reciprocity using resultants. [ABSTRACT FROM AUTHOR]

mong several implicitization methods, the method based on resultant computation is a simple and direct one, but it often brings extraneous factors which are difficult to remove. This paper studies a class of rational space curves and rational surfaces by implicitization with univariate resultant computations. This method is more efficient than the other algorithms in finding implicit equations for this class of rational curves and surfaces. [ABSTRACT FROM AUTHOR]

The n × n circulant matrix associated with the polynomial $f(t)=\sum_{i=0}^d a_i t^i$ (with d < n) is the one with first row (a0 ⋯ ad 0 ⋯ 0). The problem as to when such circulants are unimodular arises in the theory of cyclically presented groups and leads to the following question, previously studied by Odoni and Cremona: when is Res(f(t), tn-1) = ±1? We give a complete answer to this question for trinomials f(t) = tm ± tk ± 1. Our main result was conjectured by the author in an earlier paper and (with two exceptions) implies the classification of the finite Cavicchioli–Hegenbarth–Repovš generalized Fibonacci groups, thus giving an almost complete answer to a question of Bardakov and Vesnin. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we give new elementary proofs of basic results due to Gelfand, Kapranov, and Zelevinsky which express discriminants and results in terms of determinants of direct images of stably twisted Cayley–Koszul complexes of sheaves. [Copyright &y& Elsevier]

We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semicircle law and the Marčenko–Pastur law are special cases. Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form. In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (noncommutative) “free probability” theory. We hope that the tools developed allow researchers to finally harness the power of infinite random matrix theory. [ABSTRACT FROM AUTHOR]

Abstract: The computation of intersection and self-intersection loci of parametrized surfaces is an important task in Computer Aided Geometric Design. We address these problems via four resultants with separated variables; two of them are specializations of general multivariate resultants and the two others are specializations of determinantal resultants. We give a rigorous study in these four cases and provide new formulas in terms of Bezoutian matrix. [Copyright &y& Elsevier]

Abstract: A parameterized surface can be represented as a projection from a certain toric surface. This generalizes the classical homogeneous and bihomogeneous parameterizations. We extend to the toric case two methods for computing the implicit equation of such a rational parameterized surface. The first approach uses resultant matrices and gives an exact determinantal formula for the implicit equation if the parameterization has no base points. In the case the base points are isolated local complete intersections, we show that the implicit equation can still be recovered by computing any non-zero maximal minor of this matrix. The second method is the toric extension of the method of moving surfaces, and involves finding linear and quadratic relations (syzygies) among the input polynomials. When there are no base points, we show that these can be put together into a square matrix whose determinant is the implicit equation. Its extension to the case where there are base points is also explored. [Copyright &y& Elsevier]

Abstract: Explicit discriminant and resultant formulas for the Umemura polynomials associated with rational solutions to Painlevé III equation are proved; P.A. Clarkson conjectured these results, and the formulas are expressed as products of small integers to large powers, and the discriminants can also be used to determine when the roots of Umemura''s special polynomials, which have very interesting structure and striking symmetry, coalesce in the complex plane. [Copyright &y& Elsevier]

Abstract: Statistical models for genetic linkage analysis of locus diseases are -dimensional subvarieties of a -dimensional probability simplex. We determine the algebraic invariants of these models with general characteristics for ; in particular we recover, and generalize, the Hardy–Weinberg curve. For , the algebraic invariants are presented as determinants of 32×32-matrices of linear forms in nine unknowns, a suitable format for computations with numerical data. [Copyright &y& Elsevier]

Abstract: Wendt''s determinant of order n is the circulant determinant whose -th entry is the binomial coefficient , for , where n is a positive integer. We establish some congruence relations satisfied by these rational integers. Thus, if p is a prime number and k a positive integer, then and . If q is another prime, distinct from p, and h any positive integer, then . Furthermore, if p is odd, then . In particular, if , then . Also, if m and n are relatively prime positive integers, then divides . [Copyright &y& Elsevier]

We use the symmetric product of the projective line to describe the resultant scheme R n , m in as a quotient X / G where and G ⊆ Aut k ( X ) is a finite subgroup. As a special case we give a description of the discriminant scheme in terms of the symmetric product. [ABSTRACT FROM AUTHOR]

Abstract: We prove the complex analogue of Ullemar’s formula for the Jacobian of the complex moment mapping. This formula was previously established in the real case. [Copyright &y& Elsevier]

Abstract: In this paper we introduce an intermediate representation of surfaces that we call semi-implicit. We give a general definition in the language of projective complex algebraic geometry, and we begin its systematic study with an effective view-point. Our last section will apply this representation to investigate the intersection of two bi-cubic surfaces; these surfaces are widely used in Computer Aided Geometric Design. [Copyright &y& Elsevier]