query: Show that there is $z_0$ on the unit circle such that $|p(z_0)|>1$.
positive_passages.docid: 42988
positive_passages.text: Complex polynomial and the unit circle
negative_passages.docid: 90343
negative_passages.text: Proving the inequality $|e^z-1|\leq e^{|z|}-1$
negative_passages.docid: 82361
negative_passages.text: How do you prove that $p(n \xi)$ for $\xi$ irrational and $p$ a polynomial is uniformly distributed modulo 1?
negative_passages.docid: 102391
negative_passages.text: Abel summability under the hypothesis $a_n \ge 0$
negative_passages.docid: 42567
negative_passages.text: Let $(a_n)_{n \geq 0}$ be a strictly decreasing sequence of positive real numbers , and let $z \in \mathbb C$ , $|z| < 1$.
negative_passages.docid: 98662
negative_passages.text: $P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2$ for $0\le \lambda \le 1$
negative_passages.docid: 113892
negative_passages.text: Show $A \in \cal F$ exists, such that $\mathbb P(A)=1$ and $\mathbb Q(A)=0$.
negative_passages.docid: 33945
negative_passages.text: Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
negative_passages.docid: 62239
negative_passages.text: For a polynomial $p(z)$, prove there exist $R>0$, such that if $|z|=R$, then $|p(z)|\geq |a_n|R^n/2$
negative_passages.docid: 65534
negative_passages.text: Given $P(B\mid A)=1-\varepsilon$ for some $0<\varepsilon<1$ and $P(C\mid B)=1$, prove that $P(C\mid A)≥1-\varepsilon$
negative_passages.docid: 138107
negative_passages.text: Showing that $\left|\frac{z-a}{1-\bar{a}z}\right|=1$
negative_passages.docid: 149661
negative_passages.text: For $n=pq$, show that there exist $\alpha, \beta \in Z_n^*$ such that $\alpha, \beta \notin (Z_n^*)^2$ and $\alpha \cdot \beta \notin (Z_n^*)^2$.
negative_passages.docid: 106585
negative_passages.text: Show $d_{1}(x_{n},x)\rightarrow{0}$ if and only if $d_{2}(x_{n},x)\rightarrow{0}$.
negative_passages.docid: 67485
negative_passages.text: The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets
negative_passages.docid: 96116
negative_passages.text: Show $P(z) $ is a polynomial of degree $n-1$ interpolating an Analytic function.
negative_passages.docid: 114734
negative_passages.text: Prove that $\frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2+\cdots+z^{2n}|^2~dz =2n$ where $\mathcal{C}$ is the unit circle
negative_passages.docid: 14048
negative_passages.text: Proof that the complement of zero-measure set on $[0,1]$ is dense in $[0,1]$