The problem of approximating the greatest common divisor(GCD) for polynomials with inexact coefficients can be formulated as a low rank approximation problem with a Sylvester matrix. In this paper, we present an algorithm based on fast Structured Total Least Norm(STLN) for constructing a Sylvester matrix of given lower rank and obtaining the nearest perturbed polynomials with exact GCD of given...

The Extended-Row-Equivalence and Shifting (ERES) method is a matrixbased method developed for the computation of the greatest common divisor (GCD) of sets of many polynomials. In this paper we present the formulation of the shifting operation as a matrix product which allows us to study the fundamental theoretical and numerical properties of the ERES method by introducing its complete algebraic...

− The problem of finding the greatest common divisor (GCD) of univariate polynomials appears in many engineering fields. Despite its formulation is well-known, it is an ill-posed problem that entails numerous difficulties when the coefficients of the polynomials are not known with total accuracy, as, for example, when they come from measurement data. In this work we propose a novel GCD estimati...

A system that tries to analyze polyphonic musical recordings of bowed-string instruments, extract synthesis parameters of individual instrument and then re-synthesize is proposed. In the analysis part, multiple F0s estimation and partials tracking are performed based on modified WGCDV (weighted greatest common divisor and vote) method and high-order HMM. Then, dynamic time warping algorithm is ...

We discuss some problems in number theory posed by-Duro Kurepa, including the so-called left factorial hypothesis that an odd prime p does not divide 0! + 1! + · · · + (p − 1)!. 1. Introduction-D. Kurepa posed several problems in number theory that drew attention of many workers in number theory. Certainly, the most known of his problems is the so called left factorial hypothesis, which is stil...

Richard Zippel’s sparse modular GCD algorithm is widely used to compute the monic greatest common divisor (GCD) of two multivariate polynomials over Z. In this report, we present how this algorithm can be modified to solve the GCD problem for polynomials over finite fields of small cardinality. When the GCD is not monic, Zippel’s algorithm cannot be applied unless the normalization problem is r...

We propose a recursive formula for computing the remainder of a Euclidean division of polynomials (with binary coefficients), which operates in parallel on w bits at a time and takes t new incoming bits at each stage. We use this formula to design a fast parallel Cyclic Redundancy Check (CRC) system which is a look-ahead scheme that trades in arbitrary depth (processing time per cycle) and thro...

Ž . We study the depth of the ring of invariants of SL F acting on the nth 2 p symmetric power of the natural two-dimensional representation for n p. These Ž . symmetric power representations are the irreducible representations of SL F 2 p over F . We prove that, when the greatest common divisor of p y 1 and n is less p than or equal to 2, the depth of the ring of invariants is 3. We also prove...

Let K be a field. Suppose that the algebraic variety is given be the set of common solutions to a system of polynomials in n variables with coefficients in K. Given a solution P=(a_1,...,a_n) of this system with coordinates in the algebraic closure of K, we associate to it an integer called the degree of P, and defined to be the degree of the extension K(a_1,dots,a_n) over K. When all coordinat...