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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 $x_1x_2=\mathrm{const}_2$)?

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Yes, in general you need as many points as there are unknown coefficients. That this "always" works (and when it doesn't) is in fact a deeper result learned in algabraic geometry, but counerexamples involve more or less "obvious" dependencies, as e.g. in $y=b_0x_1^2+b_1x_2^2+b_2(x_1-x_2)(x_1+x_2)$, where the third summand is in fact a linear combinaton of the first two –  Hagen von Eitzen Dec 14 '12 at 11:44

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