  We address classic multivariate polynomial regression tasks from a novel
perspective resting on the notion of general polynomial $l_p$-degree, with
total, Euclidean, and maximum degree being the centre of considerations. While
ensuring stability is a theoretically known and empirically observable
limitation of any computational scheme seeking for fast function approximation,
we show that choosing Euclidean degree resists the instability phenomenon best.
Especially, for a class of analytic functions, we termed Trefethen functions,
we extend recent argumentations that suggest this result to be genuine. We
complement the novel regression scheme, presented herein, by an adaptive domain
decomposition approach that extends the stability for fast function
approximation even further.
