You don't lose information if you pick your points correctly (you store only the extrema). In the cubic case, you need the two extrema (one minimum and one maximum and you need to know whether each extremum is a min or max) and then interpolate between them.
Unfortunately this is incorrect. Extrema do not always exist (consider y=x^3) and they do not uniquely define a polynomial (y=x^2 and y=x^4 both have minima at x=0).
Unfortunately this is incorrect. Those minima are not the same. Remember that dual points have a real part and a dual part that indicates the rate of change at that point. The real part is the same but the dual is different.
My point is it’s not possible to uniquely reconstruct an arbitrary polynomial by just knowing the extrema because there may be information loss in the general case. I will stop here.
It is possible if you know the rate of change which you do with dual points. Like you don’t interpolate just position but also the dual parts I.e. rate of change.
