 

Arturas Dubickas
Polynomials with integral Mahler measures
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Published: 
June 3, 2024. 
Keywords: 
Mahler's measure, integer expanding polynomial, selfaffine 2 attractor. 
Subject [2010]: 
11R06, 11R09, 11C08, 12D10. 


Abstract
For each m in N and each sufficiently large d in N, we give an upper bound for the number of integer polynomials of degree d and Mahler's measure m. We show that there are at most exp(11(md)^{2/3}(log(md))^{4/3} of such polynomials. For `small' m, i.e. m < d^{1/2ε}, this estimate is better than the estimate m^{d(1+ε)} that comes from a corresponding upper bound on the number of integer polynomials of degree d and Mahler's measure at most m. By the results of Zaitseva and Protasov, our estimate has applications in the theory of selfaffine 2attractors. We also show that for each integer m >= 3 there is a constant c=c(m)>0 such that the number of monic integer irreducible expanding polynomials of sufficiently degree d and constant coefficient m (and hence with Mahler's measure equal to m) is at least cd^{m1}.


Acknowledgements
N/A


Author information
Arturas Dubickas
Institute of Mathematics
Faculty of Mathematics and Informatics
Vilnius University
Naugarduko 24, LT03225 Vilnius, Lithuania
arturas.dubickas@mif.vu.lt

