Let $D=\{d_1, d_2, \ldots, d_n\}$ be an integer set. I'd like to know if I can interpolate any collection of $n$ points $(x_1, y_1), (x_2, y_2), \ldots, (x_{n}, y_{n})$ by a polynomial whose degree lie in $D$. In other words, I am looking for a polynomial of the form $p(x) = \sum_{i=1}^n p_i x^{d_i}$ such that $p(x_i) = y_i$, for all $i=1, \ldots, n-1$.

Obviously, it is always possible to find such a polynomial if $D = \{0,1, \ldots, n-1\}$. On the other hand, if every element of $D$ is even, then this is not possible, since $p(x)$ would have to be even, and I can choose points $(x_i,y_i)$ breaking that. Likewise, if every element of $D$ is odd, then $p(x)$ will be odd, and interpolation is not always possible.

I'm hoping it is known exactly for which sets $D$ interpolation is possible but, barring that, any references to related literature would be appreciated.