Are interactions with quadratic terms in MARS possible? I am using the mars function of the R earth package. When the degree of interaction of MARS terms is at least 1, is it possible to have quadratic terms like x1*x1 in the model? I guess interactions between different hinge functions are possible. What about raw input variables like x1?

Yes with some modification.
On its own, MARS won't try to take higher-order functions of the predictors. MARS can only include three types of basis functions:

*

*A constant

*A hinge function

*Interactions of hinge functions

You can "trick" MARS into including a quadratic term by making a new variable xnew = x1*x1 then fitting MARS. MARS will not derive this xnew variable on its own, though.
