I am currently reading "Polynomial Methods in Combinatorics" by Prof. Larry Guth of M.I.T., where I came across a dimensional argument I couldn't quite understand.
Let $\mathbb{F}_q$ denote a field of $q$ elements, and $Poly_D(\mathbb{F}^n)$ be the space of polynomials in the ring $\mathbb{F}[x_1,...,x_n]$ with dimension $\leq{D}$ and coefficients in $\mathbb{F}^n$, such that $Poly_D(\mathbb{F}^n)\subset\mathbb{F}[x_1,...,x_n]$. Note that $Poly_D(\mathbb{F}^n)$ is a vector space over $\mathbb{F}^n_q$. We use a "Stars and Bars" encoding to compute the dimension of the space $Poly_D(\mathbb{F}^n)$. Fix $D$ and $n$. We encode the monomial $x_1^{D_1}\cdot...\cdot{x_n^{D_n}}$ by first writing $D_1$ $*$'s (meaning if $D_1=5$, we place 5 $*$'s), and placing one bar directly after. We do this until we have $D_n$ $*$'s followed directly by the $n$th bar. At the end of the encoding, if we did not max out the degree of the monomial, we add $D-\sum{D_i}\space\ast$'s.
Example: We encode $x^3y^2z^2$ in $Poly_9(\mathbb{F})$ by $\ast\ast\ast|\ast\ast|\ast\ast|\ast\ast$ .
This encoding is a bijection between the monomials of $Poly_D(\mathbb{F}^n)$ and the $\ast$'s and |'s. Clearly, we have $D+n$ "slots" where we can place our $\ast$'s and |'s. Because we have $n$ variables to place in our monomial, the number of arrangements we can have between the variables and their degrees is ${D+n}\choose{n}$.
Guth then concludes that $Dim(Poly_D(\mathbb{F}^n))={{D+n}\choose{n}}$. I'm not sure why the number of monomials is the exact dimension, any help is appreciated.