Montgomery's method of polynomial selection for the number field sieve

The number field sieve is the most efficient known algorithm for factoring large integers that are free of small prime factors. For the polynomial selection stage of the algorithm, Montgomery proposed a method of generating polynomials which relies on the construction of small modular geometric progressions. Montgomery's method is analysed in this paper and the existence of suitable geometric progressions is considered

New and Old Results in Resultant Theory

Resultants are getting increasingly important in modern theoretical physics: they appear whenever one deals with non-linear (polynomial) equations, with non-quadratic forms or with non-Gaussian integrals. Being a subject of more than three-hundred-year research, resultants are of course rather well studied: a lot of explicit formulas, beautiful properties and intriguing relationships are known in this field. We present a brief overview of these results, including both recent and already classical. Emphasis is made on explicit formulas for resultants, which could be practically useful in a future physics research.Comment: 50 pages, 15 figure

Solving Degenerate Sparse Polynomial Systems Faster

Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques allow us to sharpen and lower prior complexity bounds for this problem by fully taking into account the monomial term structure. As a corollary of our development we also obtain new explicit formulae for the exact number of isolated roots of F and the intersection multiplicity of the positive-dimensional part of Z. Finally, we present a combinatorial construction of non-degenerate polynomial systems, with specified monomial term structure and maximally many isolated roots, which may be of independent interest.Comment: This is the final journal version of math.AG/9702222 (``Toric Generalized Characteristic Polynomials''). This final version is a major revision with several new theorems, examples, and references. The prior results are also significantly improve

Symbolic-numeric algorithms for univariate polynomials

Thesis (Ph. D. in Science)--University of Tsukuba, (B), no. 2485, 2010.3.25 Includes bibliographical referencesNote to the re-typeset version: This is re-typeset version of the original dissertation. While I have maintained the original contents without changing any words and/or formulas in the main body, I have added the following information: 1. Copyright notice of corresponding articles in each chapter; 2. Digital Object Identifiers (DOI) or URLs of references as many as possible.Please note that the number of pages is slightly increased in the present edition from that of the original edition, possibly by changes of page style parameters.200

Structured matrix methods for a polynomial root solver using approximate greatest common divisor computations and approximate polynomial factorisations.

This thesis discusses the use of structure preserving matrix methods for the numerical approximation of all the zeros of a univariate polynomial in the presence of noise. In particular, a robust polynomial root solver is developed for the calculation of the multiple roots and their multiplicities, such that the knowledge of the noise level is not required. This designed root solver involves repeated approximate greatest common divisor computations and polynomial divisions, both of which are ill-posed computations. A detailed description of the implementation of this root solver is presented as the main work of this thesis. Moreover, the root solver, implemented in MATLAB using 32-bit floating point arithmetic, can be used to solve non-trivial polynomials with a great degree of accuracy in numerical examples

Blind Image Deconvolution using Approximate Greatest Common Divisor and Approximate Polynomial Factorisation

Images play a significant and important role in diverse areas of everyday modern life. Examples of the areas where the use of images is routine include medicine, forensic investigations, engineering applications and astronomical science. The procedures and methods that depend on image processing would benefit considerably from images that are free of blur. Most images are unfortunately affected by noise and blur that result from the practical limitations of image sourcing systems. The blurring and noise effects render the image less useful. An efficient method for image restoration is hence important for many applications. Restoration of true images from blurred images is the inverse of the naturally occurring problem of true image convolution through a blurring function. The deconvolution of images from blurred images is a non-trivial task. One challenge is that the computation of the mathematical function that represents the blurring process, which is known as the point spread function (PSF), is an ill-posed problem, i.e. an infinite number of solutions are possible for given inexact data. The blind image deconvolution (BID) problem is the central subject of this thesis. There are a number of approaches for solving the BID problem, including statistical methods and linear algebraic methods. The approach adopted in this research study for solving this problem falls within the class of linear algebraic methods. Polynomial linear algebra offers a way of computing the PSF size and its components without requiring any prior knowledge about the true image and the blurring PSF. This research study has developed a BID method for image restoration based on the approximate greatest common divisor (AGCD) algorithms, specifically, the approximate polynomial factorization (APF) algorithm of two polynomials. The developed method uses the Sylvester resultant matrix algorithm in the computation of the AGCD and the QR decomposition for computing the degree of the AGCD. It is shown that the AGCD is equal to the PSF and the deblurred image can be computed from the coprime polynomials. In practice, the PSF can be spatially variant or invariant. PSF spatial invariance means that the blurred image pixels are the convolution of the true image pixels and the same PSF. Some of the PSF bivariate functions, in particular, separable functions, can be further simplified as the multiplication of two univariate polynomials. This research study is focused on the invariant separable and non-separable PSF cases. The performance of state-of-the-art image restoration methods varies in terms of computational speed and accuracy. In addition, most of these methods require prior knowledge about the true image and the blurring function, which in a significant number of applications is an impractical requirement. The development of image restoration methods that require no prior knowledge about the true image and the blurring functions is hence desirable. Previous attempts at developing BID methods resulted in methods that have a robust performance against noise perturbations; however, their good performance is limited to blurring functions of small size. In addition, even for blurring functions of small size, these methods require the size of the blurring functions to be known and an estimate of the noise level to be present in the blurred image. The developed method has better performance than all the other state-of-the-art methods, in particular, it determines the correct size and coefficients of the PSF and then uses it to recover the original image. It does not require any prior knowledge about the PSF, which is a prerequisite for all the other methods

The resultant on compact Riemann surfaces

We introduce a notion of resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the Schottky double of the domain.Comment: 44 page

Cache-Friendly, Modular and Parallel Schemes For Computing Subresultant Chains

The RegularChains library in Maple offers a collection of commands for solving polynomial systems symbolically with taking advantage of the theory of regular chains. The primary goal of this thesis is algorithmic contributions, in particular, to high-performance computational schemes for subresultant chains and underlying routines to extend that of RegularChains in a C/C++ open-source library. Subresultants are one of the most fundamental tools in computer algebra. They are at the core of numerous algorithms including, but not limited to, polynomial GCD computations, polynomial system solving, and symbolic integration. When the subresultant chain of two polynomials is involved in a client procedure, not all polynomials of the chain, or not all coefficients of a given subresultant, may be needed. Based on that observation, we design so-called speculative and caching strategies which yield great performance improvements within our polynomial system solver. Our implementation of these techniques has been highly optimized. We have implemented optimized core arithmetic routines and multithreaded subresultant algorithms for univariate, bivariate and multivariate polynomials. We further examine memory access patterns and data locality for computing subresultants of multivariate polynomials, and study different optimization techniques for the fraction-free LU decomposition algorithm to compute subresultants based on determinant of Bezout matrices. Our code is publicly available at www.bpaslib.org as part of the Basic Polynomial Algebra Subprograms (BPAS) library that is mainly written in C, with concurrency support and user interfaces written in C++