Regression using Higher Order Polynomial feature

$1$ point possible $($graded$)$

Assume we have data points in the training set where is the $-$th training example:

A biochemist is considering the depicted data and we're helping them.

We want to find a non$-$linear regression function that predicts from $,$ given by

where is the polynomial feature vector that includes all and only the monomials of degree at most $($in this case, since has dimension $,$ this means has components; the degree$-$ component is redundant with the bias term $,$ but that doesn't matter for this problem$).$ What degree would you recommend the biochemist use? Note that this is a soft, not$-$completely$-$mathematical question, much like the question, $$Does Louisiana look more like a boot or a mitten?$\text{'}$ there's consensus here among folks who know the terms involved, even though it's a soft question. Common sense and human experience are important in ML engineering, so this question is fair game.