Displaying similar documents to “Subresultant Polynomial Remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x]”

A Basic Result on the Theory of Subresultants

Akritas, Alkiviadis G., Malaschonok, Gennadi I., Vigklas, Panagiotis S. (2016)

Serdica Journal of Computing

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Given the polynomials f, g ∈ Z[x] the main result of our paper, Theorem 1, establishes a direct one-to-one correspondence between the modified Euclidean and Euclidean polynomial remainder sequences (prs’s) of f, g computed in Q[x], on one hand, and the subresultant prs of f, g computed by determinant evaluations in Z[x], on the other. An important consequence of our theorem is that the signs of Euclidean and modified Euclidean prs’s - computed either in Q[x] or in Z[x] - are uniquely...

Reciprocal Stern Polynomials

A. Schinzel (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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A partial answer is given to a problem of Ulas (2011), asking when the nth Stern polynomial is reciprocal.

On the Remainders Obtained in Finding the Greatest Common Divisor of Two Polynomials

Akritas, Alkiviadis, Malaschonok, Gennadi, Vigklas, Panagiotis (2015)

Serdica Journal of Computing

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In 1917 Pell (1) and Gordon used sylvester2, Sylvester’s little known and hardly ever used matrix of 1853, to compute(2) the coefficients of a Sturmian remainder — obtained in applying in Q[x], Sturm’s algorithm on two polynomials f, g ∈ Z[x] of degree n — in terms of the determinants (3) of the corresponding submatrices of sylvester2. Thus, they solved a problem that had eluded both J. J. Sylvester, in 1853, and E. B. Van Vleck, in 1900. (4) In this paper we extend the work by Pell...

Some Algebraic Properties of Polynomial Rings

Christoph Schwarzweller, Artur Korniłowicz (2016)

Formalized Mathematics

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In this article we extend the algebraic theory of polynomial rings, formalized in Mizar [1], based on [2], [3]. After introducing constant and monic polynomials we present the canonical embedding of R into R[X] and deal with both unit and irreducible elements. We also define polynomial GCDs and show that for fields F and irreducible polynomials p the field F[X]/ is isomorphic to the field of polynomials with degree smaller than the one of p.

Real and complex pseudozero sets for polynomials with applications

Stef Graillat, Philippe Langlois (2007)

RAIRO - Theoretical Informatics and Applications

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Pseudozeros are useful to describe how perturbations of polynomial coefficients affect its zeros. We compare two types of pseudozero sets: the complex and the real pseudozero sets. These sets differ with respect to the type of perturbations. The first set – complex perturbations of a complex polynomial – has been intensively studied while the second one – real perturbations of a real polynomial – seems to have received little attention. We present a computable formula for the real...