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ON POLYNOMIALS WHOSE ROOTS HAVE RATIONAL QUOTIENT OF DIFFERENCES

Published online by Cambridge University Press:  05 July 2017

FLORIAN LUCA*
Affiliation:
School of Mathematics, University of the Witwatersrand, Private Bag X3, Wits 2050, South Africa Max Planck Institute for Mathematics, Vivatgasse 7, 53111 Bonn, Germany Department of Mathematics, Faculty of Sciences, University of Ostrava, 30. dubna 22, 701 03 Ostrava 1, Czech Republic email [email protected]
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Abstract

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We classify all polynomials $P(X)\in \mathbb{Q}[X]$ with rational coefficients having the property that the quotient $(\unicode[STIX]{x1D706}_{i}-\unicode[STIX]{x1D706}_{j})/(\unicode[STIX]{x1D706}_{k}-\unicode[STIX]{x1D706}_{\ell })$ is a rational number for all quadruples of roots $(\unicode[STIX]{x1D706}_{i},\unicode[STIX]{x1D706}_{j},\unicode[STIX]{x1D706}_{k},\unicode[STIX]{x1D706}_{\ell })$ with $\unicode[STIX]{x1D706}_{k}\neq \unicode[STIX]{x1D706}_{\ell }$.

Type
Research Article
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

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