Let $P_{0}, \ldots, P_{n}$ be a basis for the homogeneous polynomials of degree $n$ in variables $Z_{0}$ and $Z_{1}$. Then the image of the $\operatorname{map} \mathbb{P}^{1} \rightarrow \mathbb{P}^{n}$ given by

$$\left[Z_{0}, Z_{1}\right] \mapsto\left[P_{0}\left(Z_{0}, Z_{1}\right), \ldots, P_{n}\left(Z_{0}, Z_{1}\right)\right]$$

is called a rational normal curve.

Let $p_{1}, \ldots, p_{n+3}$ be a collection of points in general linear position in $\mathbb{P}^{n}$. Prove that there exists a unique rational normal curve in $\mathbb{P}^{n}$ passing through these points.

Choose a basis of homogeneous polynomials of degree 3 as above, and give generators for the homogeneous ideal of the corresponding rational normal curve.