Parametrization of integral values of polynomials

Giulio Peruginelli[1]

  • [1] Institut für Analysis und Comput. Number Theory, Technische Univ. Graz, Steyrergasse 30, A-8010 Graz, Austria.

Actes des rencontres du CIRM (2010)

  • Volume: 2, Issue: 2, page 41-49
  • ISSN: 2105-0597

Abstract

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We will recall a recent result about the classification of those polynomial in one variable with rational coefficients whose image over the integer is equal to the image of an integer coefficients polynomial in possibly many variables. These set is polynomially generated over the integers by a family of polynomials whose denominator is 2 and they have a symmetry with respect to a particular axis.We will also give a description of the linear factors of the bivariate separated polynomial f ( X ) - f ( Y ) over a number field K , which we need to formulate a conjecture for a generalization of the previous result over a generic number field.

How to cite

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Peruginelli, Giulio. "Parametrization of integral values of polynomials." Actes des rencontres du CIRM 2.2 (2010): 41-49. <http://eudml.org/doc/196289>.

@article{Peruginelli2010,
abstract = {We will recall a recent result about the classification of those polynomial in one variable with rational coefficients whose image over the integer is equal to the image of an integer coefficients polynomial in possibly many variables. These set is polynomially generated over the integers by a family of polynomials whose denominator is $2$ and they have a symmetry with respect to a particular axis.We will also give a description of the linear factors of the bivariate separated polynomial $f(X)-f(Y)$ over a number field $K$, which we need to formulate a conjecture for a generalization of the previous result over a generic number field.},
affiliation = {Institut für Analysis und Comput. Number Theory, Technische Univ. Graz, Steyrergasse 30, A-8010 Graz, Austria.},
author = {Peruginelli, Giulio},
journal = {Actes des rencontres du CIRM},
keywords = {Integer-valued polynomial; image of a polynomial; linear factor bivariate separated polynomial},
language = {eng},
number = {2},
pages = {41-49},
publisher = {CIRM},
title = {Parametrization of integral values of polynomials},
url = {http://eudml.org/doc/196289},
volume = {2},
year = {2010},
}

TY - JOUR
AU - Peruginelli, Giulio
TI - Parametrization of integral values of polynomials
JO - Actes des rencontres du CIRM
PY - 2010
PB - CIRM
VL - 2
IS - 2
SP - 41
EP - 49
AB - We will recall a recent result about the classification of those polynomial in one variable with rational coefficients whose image over the integer is equal to the image of an integer coefficients polynomial in possibly many variables. These set is polynomially generated over the integers by a family of polynomials whose denominator is $2$ and they have a symmetry with respect to a particular axis.We will also give a description of the linear factors of the bivariate separated polynomial $f(X)-f(Y)$ over a number field $K$, which we need to formulate a conjecture for a generalization of the previous result over a generic number field.
LA - eng
KW - Integer-valued polynomial; image of a polynomial; linear factor bivariate separated polynomial
UR - http://eudml.org/doc/196289
ER -

References

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  1. P.-J. Cahen and J.-L. Chabert, Integer-Valued Polynomials, Amer. Math. Soc. Surveys and Monographs, 48, Providence, 1997. Zbl0884.13010MR1421321
  2. S. Frisch, Remarks on polynomial parametrization of sets of integer points, Comm. Algebra 36 (2008), no. 3, 1110-1114. Zbl1209.11038MR2394276
  3. S. Frisch, L. Vaserstein, Parametrization of Pythagorean triples by a single triple of polynomials, Pure Appl. Algebra 212 (2008), no. 1, 271-274. Zbl1215.11025MR2355049
  4. G. Peruginelli, U. Zannier, Parametrizing over ℤ integral values of polynomials over ℚ , Comm. Algebra 38 (2010), no. 1, 119–130. Zbl1219.11048MR2597485

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