Is there a way to take a set of points $((x_0, x_1), y)$ of arbitrary length and interpolated them into a polynomial of pre-defined degree.
So we would have a function $y = f(x_0, x_1) = (a_0) + (a_1 x_0 + a_2 x_1) + (a_3 x_0 x_1 + a_4 x_0^2 + a_5 x_1^2) + degree 3 + degree 4 + ...$.
I'm interested in the case where we can pre-define the degree of the polynomial say 100. Furthermore, I'm interested in the case of Integers and even integers in some Finite Field (specifically of characteristic 2 e.g. $2^8$).
Edit
The polynomial function takes two variables as input and the output is a third variable.
I'm looking a form of Multivariant Interpolation that takes two variables but the degree of the polynomial is large e.g. degree 100.