Open Access
February 2018 Rational quotients of two linear forms in roots of a polynomial
Artūras Dubickas
Proc. Japan Acad. Ser. A Math. Sci. 94(2): 17-20 (February 2018). DOI: 10.3792/pjaa.94.17


Let $f$ and $g$ be two linear forms with non-zero rational coefficients in $k$ and $\ell$ variables, respectively. We describe all separable polynomials $P$ with the property that for any choice of (not necessarily distinct) roots $\lambda_{1},\ldots,\lambda_{k+\ell}$ of $P$ the quotient between $f(\lambda_{1},\ldots,\lambda_{k})$ and $g(\lambda_{k+1},\ldots,\lambda_{k+\ell}) \ne 0$ belongs to $\mathbf{Q}$. It turns out that each such polynomial has all of its roots in a quadratic extension of $\mathbf{Q}$. This is a continuation of a recent work of Luca who considered the case when $k=\ell=2$, $f(x_{1},x_{2})$ and $g(x_{1},x_{2})$ are both $x_{1}-x_{2}$, solved it, and raised the above problem as an open question.


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Artūras Dubickas. "Rational quotients of two linear forms in roots of a polynomial." Proc. Japan Acad. Ser. A Math. Sci. 94 (2) 17 - 20, February 2018.


Published: February 2018
First available in Project Euclid: 1 February 2018

zbMATH: 06902807
MathSciNet: MR3757092
Digital Object Identifier: 10.3792/pjaa.94.17

Primary: 11R04 , 11R11

Keywords: Conjugate algebraic numbers , quadratic extensions of $\mathbf{Q}$

Rights: Copyright © 2018 The Japan Academy

Vol.94 • No. 2 • February 2018
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