The distribution of the zeros generated by length-bounded polynomials (Q1174936)

Algebraic numbers of bounded height are separated, with the gap depending on the degrees and heights of the numbers. This paper gives estimates for the size of the hole around a rational number containing no other zeros of any integer polynomial of degree at most \(n\) and height at most \(H\). (The height of \(\sum f_ i x^ i\) is taken to be \((\sum| f_ i|^ 2)^{1/2}\). For example, for large \(n\) and \(H\), the hole around 1 contains the region \[ \{z\text{ in } \mathbb{C}\setminus\mathbb{R}:\;| z-1|<720^{1/4} n^{-5/4} H^{-1/2}\} \cup \{z \text{ in } \mathbb{R}:\;| z-1|<12^{1/2} n^{-3/2} H^{-1}\}. \] The estimates are found by picking candidates for the polynomials which will produce the neighbouring zeros. The calculations require the intervention of REDUCE. [Similar methods are used by \textit{W. Lawton}, Proc. Am. Math. Soc. 49, 47-50 (1975; Zbl 0306.12002) and \textit{L. Cerlienco}, \textit{M. Mignotte} and \textit{F. Piras}, J. Symb. Comput. 4, 21-33 (1987; Zbl 0629.12002)].