ROOTS OF UNITY AS QUOTIENTS OF TWO ROOTS OF A POLYNOMIAL
Artūras Dubickas · Journal of the Australian Mathematical Society · 2012
Abstract LetKbe a number field. Forf∈K[x], we give an upper bound on the least positive integerT=T(f) such that no quotient of two distinctTth powers of roots offis a root of unity. For each ε>0 and eachf∈ℚ[x] of degreed≥d(ε) we prove that $\log T(f)\lt (2+\varepsilon )\sqrt {d \log d}$ . In the opposite direction, we show that the constant 2 cannot be replaced by a number smaller than 1 . These estimates are useful in the study of degenerate and nondegenerate linear recurrence sequences over a number fieldK.