How would one define polynomials over the projective line $P_K^1$

Question

May $K$ be a field. If I set $\varXi=(X:Y)$ as a "projective variable" and "projective coefficients" $a_k=(x_k:y_k)\in P_K^1$ - may I then write a polynomial map $P_K^1\longrightarrow P_K^1$ in a form like

$$\sum_{k=0}^na_k\varXi^k$$

What is the form of the resulting $F(X,Y)$ ? A homogeneous polynomial ?