Complex Analysis/Global theory of holomorphic functions

Exercises
 Use Liouville's theorem to demonstrate that every non-constant polynomial $$p \in \mathbb C[z_1, \ldots, z_n]$$ has at least one root in $$\mathbb C^n$$ (Hint: Consider the function $$1/p$$). In this exercise, we want to look at the simplest sufficient conditions for the possibility of extending a function given by a real power series to a function on the complex plane.  Let $$f(x_1, \ldots, x_k) = \sum_{\alpha \in \mathbb N^k}^\infty a_\alpha (x_1, \ldots, x_k)^\alpha$$ be a power series with real coefficients which converges absolutely on an open neighbourhood of the origin of $$\mathbb R^n$$. Prove that $$f$$ may be extended to a function on an open neighbourhood of the origin of the complex plane. Let $$g(x) = \sum_{n=0}^\infty b_n x^n$$ be a power series such that for all $$n \in \mathbb N$$ $$b_n$$ is real and positive. Suppose further that $$g$$ converges for all $$x \in \mathbb R$$ st. $$|x| < R$$, where $$R > 0$$ is a real number. Prove that $$g$$ may be extended to a holomorphic function on $$B_R(0) \subset \mathbb C$$. Prove that the extensions considered in the first two sub-exercises are unique.   Let $$f: \mathbb C \to \mathbb C$$ be an entire function and let $$0 \le \alpha < 1$$, $$k \in \mathbb N$$ and $$C > 0$$ such that $$\forall z \in \mathbb C: |f(z)| \le C(1 + |z|^{k+\alpha})$$. Prove that $$f$$ is a polynomial of degree $$\le k$$. 