Topic
Irreducible polynomial
About: Irreducible polynomial is a research topic. Over the lifetime, 1680 publications have been published within this topic receiving 28195 citations.
Papers published on a yearly basis
Papers
More filters
TL;DR: This paper presents a polynomial-time algorithm to solve the following problem: given a non-zeroPolynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q (X).
Abstract: In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).
3,513 citations
TL;DR: In this paper, the authors characterize the ideals, I, for which I is CohenMacaulay in terms of topological properties of a simplicial complex associated with I. The main result is that the property of I being Cohenblacaulay, for a fixed choice of monomials, is dependent upon k (see end of Section 1 for specific examples).
373 citations
TL;DR: Probabilistic algorithms for the problems of finding an irreducible polynomial of degree n over a finite field, finding roots of a polynometric, and factoring aPolynomial into its irredUCible factors over a infinite field are presented.
Abstract: We present probabilistic algorithms for the problems of finding an irreducible polynomial of degree n over a finite field, finding roots of a polynomial, and factoring a polynomial into its irreducible factors over a finite field. All of these problems are of importance in algebraic coding theory, algebraic symbol manipulation, and number theory. These algorithms have a very transparent, easy to program structure. For finite fields of large characteristic p, so that exhaustive search through ${\text{Z}}_p $, is not feasible, our algorithms are of lower order in the degrees of the polynomial and fields in question, than previously published algorithms.
369 citations
329 citations
TL;DR: A new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials are introduced, and a new codification of the set of solutions of a positive dimensional algebraic variety is given relying on a new global version of Newton's iterator.
323 citations