If we're working with several number of features, then we deal with a high dimensional space. I found out that in particular he number of my cofficients is $O(D^M)$ where $D$ is the number of features and $M$ the complexity of the model (degree of polynomial), but when I learned about linear models we have that
$$y(x,\textbf{w}) = w_0 + w_1x + w_2x^2+ \dots + w_Mx^M = \sum_{i=0}^M w_ix^i$$
so in this case the number of coefficients grows linearly with model complexity (the degree of polynomial).
I'm thinking that I may confuse the degree of polynomial with the number of features since every monoid is gonna be $D$-dimensonal..
If I consider $\textbf{x} \in \mathbb{R}^D$ how can I properly explain $y(\textbf{x,w})$ where I want to express both the dimension and degree of polynomial ad why?