论文标题
单位圆上的零多项式：Leja序列的规律性
Polynomials with Zeros on the Unit Circle: Regularity of Leja Sequences
论文作者
论文摘要
令$ z_1，\ dots，z_m $ be $ m $不同的复数数字，标准化为$ | z_k | = 1 $，并考虑多项式$$ p_ {m}（z）= \ prod_ {k = 1}^{m} {（z -z_k）}。 \ mbox {where} 〜z^* = \ arg \ max_ {| z | = 1} | p_ {n}（z）|，$$，并证明，与初始多项式$ p_m $，$ p_ {n} $ equidistribute的根部无关。当有时手工添加“对抗性”点时，这甚至可以持续。我们根据涉及涉及反向分数laplacian $（ - δ）^{ - 1/2} $的动态系统的主要结果重新塑造，并猜测是，当用这种语言用短语时，基本的规律性现象可能会以非常一般的环境出现。
Let $z_1, \dots, z_m$ be $m$ distinct complex numbers, normalized to $|z_k| = 1$, and consider the polynomial $$ p_{m}(z) = \prod_{k=1}^{m}{(z-z_k)}.$$ We define a sequence of polynomials in a greedy fashion, $$ p_{N+1}(z) = p_{N}(z) \left(z - z^*\right)\qquad \mbox{where}~z^* = \arg\max_{|z|=1} |p_{N}(z)|,$$ and prove that, independently of the initial polynomial $p_m$, the roots of $p_{N}$ equidistribute in angle at rate at most $(\log{N})^2/N$. This even persists when sometimes adding `adversarial' points by hand. We rephrase the main result in terms of a dynamical system involving the inverse fractional Laplacian $(-Δ)^{-1/2}$ and conjecture that, when phrased in this language, the underlying regularity phenomenon might appear in a very general setting.
