Show that on local charts, a function $f : \mathbb{P}^n \rightarrow L_d$ is defined by polynomials.

Question

Let $$L_d = (\mathbb{C}^{n+1}−\{0\})×\mathbb{C}/∼, \text{ where } (x_0,\dots,x_n,q)∼(λx_0,…λx_n,λ^dq).$$ Let $w(x_0, \dots, x_n)$ be a homogeneous polynomial of degree $d.$ I've already shown that $$f:\mathbb{P}^n \rightarrow L^d$$ defined by $$[x_0:x_1:\dots:x_n] \mapsto [x_0:x_1:\dots : x_n : w(x_0, \dots, x_n)]$$ is a well-defined function. My goal is to show that on local charts it is defined by polynomials. I know that the open charts on $L_d$ are $$U_i = \{[x_0: \dots: x_n: q] | x_i \neq 0\}$$ and the open charts on $\mathbb{P}^n$ are $$U_i = \{[x_0: \dots: x_n] | x_i \neq 0\}.$$ Are local charts the same thing as open charts? What does it mean for the function to be defined by polynomials?