论文标题
相互多项式的非管根的数量
The number of nonunimodular roots of a reciprocal polynomial
论文作者
论文摘要
我们引入了一个元素倒数多项式的序列P_D，其整数系数具有固定的中心系数以及外围系数。我们证明，当d倾向于无穷大时，P_D的非管根数量与其D度d的比率具有限制。我们表明，如果多项式的系数在模量中可以任意较大，那么L可以任意接近0。似乎合理地相信，如果系数有限，那么Lehmer的猜想的类似物是真实的：L = 0是真实的，或者是否存在一个差距，我们就无法任意限制A值。我们估计了多项式家族的极限比。我们计算了多项式与许多具有小Mahler度量的双变量多项式相关的多项式的极限比。
We introduce a sequence P_d of monic reciprocal polynomials with integer coefficients having the central coefficients fixed as well as the peripheral coefficients. We prove that the ratio between number of nonunimodular roots of P_d and its degree d has a limit L when d tends to infinity. We show that if the coefficients of a polynomial can be arbitrarily large in modulus then L can be arbitrarily close to 0. It seems reasonable to believe that if the coefficients are bounded then the analogue of Lehmer's Conjecture is true: either L = 0 or there exists a gap so that L could not be arbitrarily close to 0. We present an algorithm for calculation the limit ratio and a numerical method for its approximation. We estimated the limit ratio for a family of polynomials. We calculated the limit ratio of polynomials correlated to many bivariate polynomials having small Mahler measure.
