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Computing Roots Of Polynomials Over Function Fields Of Curves
, 1998
"... . We design algorithms for finding roots of polynomials over function fields of curves. Such algorithms are useful for list decoding of ReedSolomon and algebraicgeometric codes. In the first half of the paper we will focus on bivariate polynomials, i.e., polynomials over the coordinate ring of the ..."
Abstract

Cited by 22 (3 self)
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. We design algorithms for finding roots of polynomials over function fields of curves. Such algorithms are useful for list decoding of ReedSolomon and algebraicgeometric codes. In the first half of the paper we will focus on bivariate polynomials, i.e., polynomials over the coordinate ring of the affine line. In the second half we will design algorithms for computing roots of polynomials over the function field of a nonsingular absolutely irreducible plane algebraic curve. Several examples are included. 1. Introduction In this paper we will study the following problem: given a nonsingular absolutely irreducible plane curve X over the finite field F q , a divisor G on X , and a polynomial H defined over the function field of X , compute all zeros of H that belong to L(G). Our interest in this problem stems mainly from recent list decoding algorithms [5, 9, 11] for ReedSolomon and algebraic geometric codes. Originally, those algorithms found the roots of H by completely factoring i...
Tests and Constructions of Irreducible Polynomials over Finite Fields
 In Foundations of Computational Mathematics
, 1997
"... In this paper we focus on tests and constructions of irreducible polynomials over finite fields. We revisit Rabin's [1980] algorithm providing a variant of it that improves Rabin's cost estimate by a log n factor. We give a precise analysis of the probability that a random polynomial of de ..."
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Cited by 12 (4 self)
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In this paper we focus on tests and constructions of irreducible polynomials over finite fields. We revisit Rabin's [1980] algorithm providing a variant of it that improves Rabin's cost estimate by a log n factor. We give a precise analysis of the probability that a random polynomial of degree n contains no irreducible factors of degree less than O(log n). This probability is naturally related to BenOr's [1981] algorithm for testing irreducibility of polynomials over finite fields. We also compute the probability of a polynomial being irreducible when it has no irreducible factors of low degree. This probability is useful in the analysis of various algorithms for factoring polynomials over finite fields.