What is the lowest degree to guarantee a polynomial exists which go through $N$ points?

Question

Suppose I have $N$ points $(x_i, y_i)$ on $\mathbb{R}^2$. What is $D$ such that we can say there exists a degree $D$ polynomial $p$ with real coefficients satisfying $y_i = p (x_i)$ for each of the $N$ points? How do we prove it? Thank you!

Answer 1

Degree $N-1$ will do the job, as long as the $x_i$ are distinct. This generalizes "two points determine a line."

See Lagrange interpolation.