Nonreciprocal algebraic numbers of small Mahler's measure

Artūras Dubickas ¹   ; Jonas Jankauskas ¹

¹ Department of Mathematics and Informatics Vilnius University Naugarduko 24 Vilnius LT-03225, Lithuania

Acta Arithmetica, Tome 157 (2013) no. 4, pp. 357-364

Résumé

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 $d$.

Détail

DOI : 10.4064/aa157-4-3

Keywords: prove there exist least monic irreducible nonreciprocal polynomials integer coefficients degree whose mahler measures smaller where absolute positive constant these polynomials constructed nonreciprocal divisors newman hexanomials cdots where integers leq cdots leq satisfy restrictions including result improves previous lower bound seems closer correct value function best known upper bound which exponential nbsp