Displaying similar documents to “The field of Nash functions and factorization of polynomials”

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.

Transcendence results on the generating functions of the characteristic functions of certain self-generating sets, II

Peter Bundschuh, Keijo Väänänen (2015)

Acta Arithmetica

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This article continues a previous paper by the authors. Here and there, the two power series F(z) and G(z), first introduced by Dilcher and Stolarsky and related to the so-called Stern polynomials, are studied analytically and arithmetically. More precisely, it is shown that the function field ℂ(z)(F(z),F(z⁴),G(z),G(z⁴)) has transcendence degree 3 over ℂ(z). This main result contains the algebraic independence over ℂ(z) of G(z) and G(z⁴), as well as that of F(z) and F(z⁴). The first...

Mean value theorems for L-functions over prime polynomials for the rational function field

Julio C. Andrade, Jonathan P. Keating (2013)

Acta Arithmetica

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The first and second moments are established for the family of quadratic Dirichlet L-functions over the rational function field at the central point s=1/2, where the character χ is defined by the Legendre symbol for polynomials over finite fields and runs over all monic irreducible polynomials P of a given odd degree. Asymptotic formulae are derived for fixed finite fields when the degree of P is large. The first moment obtained here is the function field analogue of a result due to...

On Roots of Polynomials and Algebraically Closed Fields

Christoph Schwarzweller (2017)

Formalized Mathematics

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In this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].

Algebraic condition for decomposition of large-scale linear dynamic systems

Henryk Górecki (2009)

International Journal of Applied Mathematics and Computer Science

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The paper concerns the problem of decomposition of a large-scale linear dynamic system into two subsystems. An equivalent problem is to split the characteristic polynomial of the original system into two polynomials of lower degrees. Conditions are found concerning the coefficients of the original polynomial which must be fulfilled for its factorization. It is proved that knowledge of only one of the symmetric functions of those polynomials of lower degrees is sufficient for factorization...

Differentiability of Polynomials over Reals

Artur Korniłowicz (2017)

Formalized Mathematics

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In this article, we formalize in the Mizar system [3] the notion of the derivative of polynomials over the field of real numbers [4]. To define it, we use the derivative of functions between reals and reals [9].

Comments on the height reducing property

Shigeki Akiyama, Toufik Zaimi (2013)

Open Mathematics

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A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring ℤ of the rational integers such that ℤ[α] = F[α]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one,...