I have a question related to interaction in polynomial regression. So, suppose I have the following model:

Y = aX1 + bX2 + c*X1^2 + d*X2^2 + e*X1*X2 + f*K + error,

where f is a vector of coefficients, and K is a vector of variables that may have interaction terms between them.

Now, as I understand, one should include the X1*X2 interaction term, since we have both X1^2 and X2^2 in the model. However, we can see that there is no interaction between X1 and X2^2 or between X2 and X1^2.

My question is: If we have a similar model, such as:

Y = aX1 + bX2 + c*X1^2 + e*X1*X2 + f*K + error,

in this context, do we have to include the term X2*X1^2 in the model? If so, why?

Thank you in advance!,

Tux