Nonreciprocal algebraic numbers of small Mahler's measure
Volume 157 / 2013
Acta Arithmetica 157 (2013), 357-364
MSC: Primary 11R06; Secondary 11R09.
DOI: 10.4064/aa157-4-3
Abstract
We prove that there exist at least $cd^5$ monic irreducible nonreciprocal polynomials with integer coefficients of degree at most $d$ whose Mahler measures are smaller than $2$, where $c$ is some absolute positive constant. These polynomials are constructed as nonreciprocal divisors of some Newman hexanomials $1+x^{r_1}+\cdots +x^{r_5}$, where the integers $1 \leq r_1< \cdots < r_5 \leq d$ satisfy some restrictions including $2r_j< r_{j+1}$ for $j=1,2,3,4$. This result improves the previous lower bound $cd^3$ and seems to be closer to the correct value of this function in $d$ than the best known upper bound which is exponential in $d$.
