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

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

Zbl 1435.11061
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References:

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