反多项式的非幺模根的个数
The number of nonunimodular roots of a reciprocal polynomial
摘要：
本文引入了一个整系数单倒数整系数多项式序列P_d，它的中心系数和外围系数都是固定的.证明了当d趋于无穷大时，P_d的非幺模根数与其次数d之比存在极限L.我们表明，如果一个多项式的系数可以是任意大的模，那么L可以任意接近0。我们似乎有理由相信，如果系数是有界的，那么类似的莱默猜想是正确的：要么L = 0，要么存在一个间隙，使L不能任意接近0。给出了计算极限比的一种算法及其近似的数值方法。我们估计了一族多项式的极限比。我们计算了与许多具有小马勒测度的二元多项式相关的多项式的极限比。
英文摘要：
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.
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