A $d$-form on ${\mathbb R}^n$ that vanishes on $\binom{d+n-1}{n-1}$ general points, vanishes identically.

Question

I'm looking for a reference for the fact that a $d$-form on ${\mathbb R}^n$ that vanishes on $p_1,..,p_{\binom{d+n-1}{n-1}}$ general points, vanishes identically. A specific construction of a set of points is also welcomed. Thank you.

Answer 1

The dimension of the vector space of homogeneous polynomials $R_d(\mathbb R^n)$ is indeed $\binom{d+n-1}{n-1}$.

But for $n>1$ your claim does not hold for all layouts of the points. (It's a consequence of Mairhuber–Curtis theorem proving that there are no Haar spaces in $n>1$).

For example, take a linear $d=1$ polynomial $ax+by\in R_1$ on $\mathbb R^2$. The graph of $p=ax+by$, a plane, intersects the $xy$-plane along the line $ax+by=0$, so if the both ($\binom{d+n-1}{n-1}=2$) of our sample points are on the intersection line then $p=0$, but not necessarily $p\equiv0$. For the latter we need to take the second point to be not collinear with the first and the origin. Demanding that $p=0$ on these two points (and the origin) then leads to $p\equiv0$. Such arrangements of points are called unisolvent, at least in the field of approximation theory and multivariate polynomial interpolation. See, for example, the Definition 2.6 in Wendland's "Scattered data approximation". It basically says

Definition. Points are called $\Pi$-unisolvent if the zero polynomial is the only polynomial from $\Pi$ that vanishes on all of them.