query: Prove that an entire function with modulus bounded from below by a polynomial is a polynomial itself.
positive_passages.docid: 146506
positive_passages.text: If a complex function $f$ is Entire, and there exist $k,R > 0$, $n \in N$ such that $|f(z)| > k|z|^n$ for all $|z| > R$, then f is a polynomial.
negative_passages.docid: 147047
negative_passages.text: Does there exist a polynomial $p(x) \in \mathbb C[x]$ such that $p(x) \notin \mathbb R[x]$ and $p(x)p(-x)=p(x^2)$?
negative_passages.docid: 55119
negative_passages.text: Proof that the coefficients of a polynomial are real
negative_passages.docid: 84697
negative_passages.text: Show that if $|f(z)| \leq M |z|^n$ then $f$ is a polynomial max degree n
negative_passages.docid: 79359
negative_passages.text: Show that $f$ is a polynomial if it's the uniform limit of polynomais
negative_passages.docid: 33011
negative_passages.text: Prove that the polynomial $f_n(x)=nx^{n+1}-(n+1)x^n+1$ is divisible by $(x-1)^2$
negative_passages.docid: 127547
negative_passages.text: Prove that If $f$ is polynomial function of even degree $n$ with always $f\geq0$ then $f+f'+f''+\cdots+f^{(n)}\geq 0$.
negative_passages.docid: 155444
negative_passages.text: Show that entire function $f$ is a polynomial of degree at most $n$
negative_passages.docid: 36379
negative_passages.text: Suppose that $ f $ is entire and that for each $ z $, either $ |f(z)| \leq 1 $ or $ |f^\prime (z) |\leq 1 $. Prove that $ f $ is a linear polynomial.
negative_passages.docid: 69804
negative_passages.text: Prove that if $f$ is entire and $\vert f(z^2) \vert \leq 2 \vert f(z) \vert$, then $f$ is constant
negative_passages.docid: 6722
negative_passages.text: to show $f$ is a polynomial in $z$
negative_passages.docid: 131289
negative_passages.text: Polynomial with a root modulo every prime but not in $\mathbb{Q}$.
negative_passages.docid: 97579
negative_passages.text: showing a function defined from an integral is entire
negative_passages.docid: 141184
negative_passages.text: Proof that a degree 4 polynomial has at least two roots
negative_passages.docid: 13981
negative_passages.text: Proof that every polynomial of odd degree has one real root
negative_passages.docid: 105172
negative_passages.text: Prove that the entire function is constant over certain conditions.
negative_passages.docid: 103932
negative_passages.text: How can I prove that a polynomial with degree $n$ is continuous everywhere in $\mathbb{R}$ using definitions?