Studies On The Smallest Value Of The Maximal Modulus Of A Reciprocal Algebraic Integer

The Mahler measure of a polynomial P is the absolute value of the product of all roots of P that have modulus at least 1 multiplied by the leading coefficient. The maximal modulus of an algebraic integer which is related to Mahler measure is an open question in computational number theory. Maximal modulus of an algebraic integer is the largest modulus of its conjugates, also called the house of an algebraic integer which is denoted by (?). Let a be an algebraic integer of degree d, with conjugatesα₁=α₁,α₂,…α_d,then(?)=max_{1≤i≤d}|α_i|.An algebraic integer is reciprocal if its minimal polynomial is reciprocal. Letαbe a reciprocal algebraic integer, deg(α) = 2d andP=b₀x^2d+b₁x^2d-1+…+b_2d-1x+b_2d=Multiply from i=1 to 2d(x-α_i)is its minimal polynomial, where b₀=b_2d=1?b_i=b_2d-i.We define s₁=sum from i=1 to 2dα_i,s_k=sum from i=1 to 2dα_i^k.For the smallest houses of reciprocal algebraic integers, Boyd[l] computed the smallest houses of reciprocal algebraic integers for d≤16. He used the following method: for a fixed degree d and given B, there exists algebraic integerαsuch that (?)≤B,where deg(α) = d. Obviously,|s_k|≤dB^k if (?)≤B. Then he used these bounds and Newton's formula:s_k+s_k-1b₁+…+s₁b_k-1+kb_k=0which gives by induction bounds for the coefficients b_k for 1≤k≤d to get a large set F_d of reciprocal polynomials where we find the smallest house. We follow Boyd's strategy, but the computing time grows exponentially with degree d. In this paper, we give new better bounds for s_k using a large family of explicit auxiliary functions with the following type:f(z)=-Re(z)-sum from j=1 to J e_j log|Q_j(z)|.This method has been used in [10]. With this method, we compute the smallest houses of reciprocal algebraic integers for 2≤d≤26[19].