I have a question on multiple polynomial regression and the absolute minimum amount of points in the different terms. The minimum amount of points required for a second order polynomial would (in one variable) be three and in general it would $p+1$, $p$ being the polynomial order. I have the intuition this generalizes to more than one variable but I cannot prove it and I would like to inspect my design matrix term by term (thus column by column) for sufficient amounts of points. For example take the following model: $ y=b_0+b_1x_1+b_2x_2+b_{12}x_1x_2 $ The order of the last term is $2$, suggesting three points are needed to sufficiently vary this term. Can anyone guide me to a proof that the third column in my design matrix should have at least three distinct values?
Another way of stating the question would be: would one run into trouble sampling the points $x_1$ and $x_2$ on the hyperbola $x_1x_2=\mathrm{const}_1$ (or on $x_1x_2=\mathrm{const}_1$ and $x_1x_2=\mathrm{const}_2$)?