Exact resultants for corner-cut unmixed multivariate polynomial systems using the dixon formulation

Structural conditions on the support of a multivariate polynomial system are developed for which the Dixon-based resultant methods compute exact resultants. For cases when this cannot be done, an upper bound on the degree of the extraneous factor in the projection operator can be determined a priori, thus resulting in quick identification of the extraneous factor in the projection operator. (For the bivariate case, the degree of the extraneous factor in a projection operator can be determined a priori.) The concepts of a corner-cut support and almost corner-cut support of an unmixed polynomial system are introduced. For generic unmixed polynomial systems with corner-cut and almost corner-cut supports, the Dixon based methods can be used to compute their resultants exactly. These structural conditions on supports are based on analyzing how such supports differ from box supports of n-degree systems for which the Dixon formulation is known to compute the resultants exactly. Such an analysis also gives a sharper bound on the complexity of resultant computation using the Dixon formulation in terms of the support and the mixed volume of the Newton polytope of the support. These results are a direct generalization of the authors ’ results on bivariate systems including the results of Zhang and Goldman as well as of Chionh for generic unmixed bivariate polynomial systems with corner-cut supports

The Resultant of an Unmixed Bivariate System

This paper gives an explicit method for computing the resultant of any sparse unmixed bivariate system with given support. We construct square matrices whose determinant is exactly the resultant. The matrices constructed are of hybrid Sylvester and B\'ezout type. We make use of the exterior algebra techniques of Eisenbud, Fl{\o}ystad, and Schreyer.Comment: 18 pages, 2 figure

Rational Univariate Reduction via toric resultants

AbstractWe describe algorithms for solving a given system of multivariate polynomial equations via the Rational Univariate Reduction (RUR). We compute the RUR from the toric resultant of the input system. Our algorithms derandomize several of the choices made in similar prior algorithms. We also propose a new derandomized algorithm for solving an overdetermined system. Finally, we analyze the computational complexity of the algorithm, and discuss its implementation and performance

The Canny–Emiris Conjecture for the Sparse Resultant

We present a product formula for the initial parts of the sparse resultant associated with an arbitrary family of supports, generalizing a previous result by Sturmfels. This allows to compute the homogeneities and degrees of this sparse resultant, and its evaluation at systems of Laurent polynomials with smaller supports. We obtain an analogous product formula for some of the initial parts of the principal minors of the Sylvester-type square matrix associated with a mixed subdivision of a polytope. Applying these results, we prove that under suitable hypothesis, the sparse resultant can be computed as the quotient of the determinant of such a square matrix by one of its principal minors. This generalizes the classical Macaulay formula for the homogeneous resultant and confirms a conjecture of Canny and Emiris.Fil: D'Andrea, Carlos. Centre de Recerca Matemàtica; España. Universidad de Barcelona; EspañaFil: Jeronimo, Gabriela Tali. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Sombra, Martín. Centre de Recerca Matemàtica; España. Institució Catalana de Recerca I Estudis Avançats; España. Universidad de Barcelona; Españ

Algorithmic Contributions to the Theory of Regular Chains

Regular chains, introduced about twenty years ago, have emerged as one of the major tools for solving polynomial systems symbolically. In this thesis, we focus on different algorithmic aspects of the theory of regular chains, from theoretical questions to high- performance implementation issues. The inclusion test for saturated ideals is a fundamental problem in this theory. By studying the primitivity of regular chains, we show that a regular chain generates its saturated ideal if and only if it is primitive. As a result, a family of inclusion tests can be detected very efficiently. The algorithm to compute the regular GCDs of two polynomials modulo a regular chain is one of the key routines in the various triangular decomposition algorithms. By revisiting relations between subresultants and GCDs, we proposed a novel bottom-up algorithm for this task, which improves the previous algorithm in a significant manner and creates opportunities for parallel execution. This thesis also discusses the accelerations towards fast Fourier transform (FFT) over finite fields and FFT based subresultant chain constructions in the context of massively parallel GPU architectures, which speedup our algorithms by several orders of magnitude

Exact polynomial system solving for robust geometric computation

I describe an exact method for computing roots of a system of multivariate polynomials with rational coefficients, called the rational univariate reduction. This method enables performance of exact algebraic computation of coordinates of the roots of polynomials. In computational geometry, curves, surfaces and points are described as polynomials and their intersections. Thus, exact computation of the roots of polynomials allows the development and implementation of robust geometric algorithms. I describe applications in robust geometric modeling. In particular, I show a new method, called numerical perturbation scheme, that can be used successfully to detect and handle degenerate configurations appearing in boundary evaluation problems. I develop a derandomized version of the algorithm for computing the rational univariate reduction for a square system of multivariate polynomials and a new algorithm for a non-square system. I show how to perform exact computation over algebraic points obtained by the rational univariate reduction. I give a formal description of numerical perturbation scheme and its implementation

Macroinstability and Perturbation in Turbulent Stirred Tank Flows

Impeller stirred tank reactors (STRs) are commonly used in the chemical processing industries for a variety of mixing and blending technologies. In this research, a numerical study of flow and mixing inside turbulently agitated STRs are carried out. An immersed boundary method (IBM) is utilized to represent moving impeller geometries in the background of multi-block structured curvilinear fluid. The IBM This curvilinear-IBM methodology is further combined with the large eddy simulation (LES) technique to address the issue of modeling unsteady turbulent flows in the STR. Verification of the combined IBM-LES implementation strategy in curvilinear coordinates is done through comparisons with the measurements of laminar and turbulent flows in baffled STRs with pitched blade impellers. Flow structures are studied inside a dished bottom pitched-blade baffled for different impeller rotational speeds in the turbulent regime to observe the formation of trailing edge vortices which are associated with higher levels of turbulent kinetic energy relative to the remaining flowfield. Instabilities occurring at a frequency lower than the frequency of impeller rotation are identified from the time signal of velocity components. The role of these low frequency macro-instabilities (MI) is explored by observing changes in the three-dimensional circulation pattern within the STR. Significant amount of kinetic energy is observed to be associated with the dynamics of the trailing edge vortices during MI cycles. Flow inside an unbaffled Rushton impeller STR is perturbed using time-dependent impeller rotational speeds at a dominant MI frequency. Perturbation increased the mean radial width of the impeller jet-stream and enhanced overall turbulent kinetic energy compared to the constant rotational speed cases. Large-scale periodic velocity fluctuations due to perturbations produced large strain rates favoring higher turbulence production. Fluctuations in power consumptions are shown to correlate with the perturbation amplitude. Study on the mixing of a passive scalar inside STR showed that the growth rate of unmixed tracer is influenced by the MI oscillations. Perturbation of the STR flow resulted into significant reduction of mixing time. The spatio-temporal behavior of the large-scale mixing structures revealed that fast mixing is promoted due to the break-up of unmixed segregated zones during a perturbation cycle

Lectures on Computational Numerical Analysis of Partial Differential Equations

From Chapter 1: The purpose of these lectures is to present a set of straightforward numerical methods with applicability to essentially any problem associated with a partial differential equation (PDE) or system of PDEs independent of type, spatial dimension or form of nonlinearity.https://uknowledge.uky.edu/me_textbooks/1002/thumbnail.jp