Multivariable Polynomial Regression I want to regress my data into a function as: $a0+a_1K+a_2K^2+a_3T+a_4T^2+a_5KT$. But setting $\mathbf{a}=(\mathbf{K^\top K})^{-1} \mathbf{K^\top Y}$ does not work, I think the reason why is since the K and T elements of the matrix $\mathbf{K}$ are linearly dependent. Is there a way/ways to approximate the aforementioned function into my data?

From your comment, it looks like $T = \alpha K$, i.e., you have a multicolinearity in your data. Hence, you don't need both $K$ and $T$ - you should get rid of one of them.