Why is the use of high order polynomials for regression discouraged? I've read many times on this site that high order polynomials (generally more than third) shouldn't be used in linear regression, unless there is a substantial justification to do so.
I understand the issues about extrapolation (and prediction at the boundaries).
Since extrapolation isn't important to me...

*

*Are high order polynomials also a bad way of approximating the underlying function within the range of the data points? (i.e. interpolation)

*If so, what problems are arising?

I don't mind being redirected to a good book or paper about this.
Thanks.

If your goal is interpolation, you typically want the simplest function that describes your observations and avoid overfitting.
Given that it is unusual to see physical laws and relationships which contain powers higher than 3, using higher order polynomials when there is no intuitive justification is taken to be a sign of overfitting.
See also this similar question.
