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Nonreciprocal algebraic numbers of small Mahler's measure

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Dubickas A., Jankauskas J.

Acta Arithmetica

2013

tom 157

nr 4

357-364

We prove that there exist at least cd⁵ 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 $1 + x^{r₁} + ⋯ + x^{r₅}$, where the integers 1 ≤ r₁ < ⋯ < r₅ ≤ d satisfy some restrictions including $2r_{j} < r_{j+1}$ for j = 1,2,3,4. This result improves the previous lower bound cd³ 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.

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1 Acta Arithmetica

1 Dubickas A.

1 Jankauskas J.

1 2013

Wyniki wyszukiwania

Nonreciprocal algebraic numbers of small Mahler's measure