Rational quotients of two linear forms in roots of a polynomial. (English) Zbl 1459.11080

Dubickas, Artūras

Proc. Japan Acad., Ser. A 94, No. 2, 17-20 (2018).

Summary: 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 \(g(\lambda_{k+1},\ldots,\lambda_{k+\ell}) \neq 0\) belongs to \(\mathbb{Q}\). It turns out that each such polynomial has all of its roots in a quadratic extension of \(\mathbb{Q}\). This is a continuation of a recent work of F. Luca [Bull. Aust. Math. Soc. 96, No. 2, 185–190 (2017; Zbl 1435.11061)] 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.

MSC:

11C08	Polynomials in number theory
11R04	Algebraic numbers; rings of algebraic integers
11R11	Quadratic extensions

Keywords:

conjugate algebraic numbers; quadratic extensions of \(\mathbb{Q}\)

Citations:

Zbl 1435.11061

References:

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