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So I can't seem to find much on this by searching so I came here. Let's say I had 3 variables $x_1,x_2,x_3$ and the let's say the degree of the polynomial was $d=2$, I can define the length of a vector of variables as $l$ such that $l=3$ in this case. What I'm looking for is the length of a vector of coefficients that would be needed to create a polynomial regression on these variables.

Would I be right in saying that the length of the vectors of coefficients needed for polynomial regression would be $d\times l$?

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Yes - it is $d\times l$. There would be 6 $β$ coefficients for 2*3:

$β_1x_1 + β_2x_1^2 + β_3x_2 + β_4x_2^2 + β_5x_3 + β_6x_3^2$

That does not include an intercept or any interaction terms.

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