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.


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


Zbl 1435.11061
Full Text: DOI Euclid


[1] A. Dubickas, On the degree of a linear form in conjugates of an algebraic number, Illinois J. Math. 46 (2002), no. 2, 571-585. · Zbl 1028.11066
[2] A. Dubickas and C. J. Smyth, Variations on the theme of Hilbert’s Theorem 90, Glasg. Math. J. 44 (2002), no. 3, 435-441. · Zbl 1112.11308
[3] A. Dubickas, Additive Hilbert’s Theorem 90 in the ring of algebraic integers, Indag. Math. (N.S.) 17 (2006), no. 1, 31-36. · Zbl 1112.11050
[4] F. Luca, On polynomials whose roots have rational quotient of differences, Bull. Aust. Math. Soc. 96 (2017), no. 2, 185-190. · Zbl 1435.11061
[5] N. Saxena, S. Severini and I. E. Shparlinski, Parameters of integral circulant graphs and periodic quantum dynamics, Int. J. Quantum Inf. 5 (2007), 417-430. · Zbl 1119.81042
[6] T. Zaïmi, On the integer form of the additive Hilbert’s Theorem 90, Linear Algebra Appl. 390 (2004), 175-181.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.