## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Rational quotients of two linear forms in roots of a polynomial

Artūras Dubickas

#### Abstract

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.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 94, Number 2 (2018), 17-20.

Dates
First available in Project Euclid: 1 February 2018

https://projecteuclid.org/euclid.pja/1517454034

Digital Object Identifier
doi:10.3792/pjaa.94.17

#### Citation

Dubickas, Artūras. Rational quotients of two linear forms in roots of a polynomial. Proc. Japan Acad. Ser. A Math. Sci. 94 (2018), no. 2, 17--20. doi:10.3792/pjaa.94.17. https://projecteuclid.org/euclid.pja/1517454034

#### References

• A. Dubickas, On the degree of a linear form in conjugates of an algebraic number, Illinois J. Math. 46 (2002), no. 2, 571–585.
• A. Dubickas and C. J. Smyth, Variations on the theme of Hilbert's Theorem 90, Glasg. Math. J. 44 (2002), no. 3, 435–441.
• A. Dubickas, Additive Hilbert's Theorem 90 in the ring of algebraic integers, Indag. Math. (N.S.) 17 (2006), no. 1, 31–36.
• P. Habegger, The norm of Gaussian periods. (to appear in Q. J. Math.).
• F. Luca, On polynomials whose roots have rational quotient of differences, Bull. Aust. Math. Soc. 96 (2017), no. 2, 185–190.
• 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.
• T. Zaïmi, On the integer form of the additive Hilbert's Theorem 90, Linear Algebra Appl. 390 (2004), 175–181.