1D interpolation: finding a polynomial satisfying $\forall_i\ p(x_i)=y_i$ can be written as a system of linear equations, having well known Vandermonde determinant: $\det=\prod_{i<j} (x_i-x_j)$. Hence, the interpolation problem is well defined as long as the system of equations is determined ($\det\neq 0$), that is equivalent with condition of having no two repeating $x$-s.
I need something analogous for quadratic form in $n$ dimensions: we would like to find symmetric matrix $A$ satisfying $\forall_k\ f(x^k)=y_k$, where $f(x)=x^T A x$, this time $x^k$ are vectors.
We get a system of linear equations: $$ \forall_k\ \sum_i A_{ii} (x^k_i)^2 + 2\sum_{i<j} A_{ij} x_i^k x_j^k =y_k$$ for $D=n(n+1)/2$ coefficients of symmetric $A$.
In analogy to interpolation problem, having values in $D$ points, we would like to find $A$. However, it requires that $\det\neq 0$ for the above set of linear equations.
Is there known a compact form for this determinant? (in analogy to Vandermonde)
If not, are there some known conditions ensuring it is nonzero - making fitting quadratic form well defined? These conditions need to contain e.g. that no two points are in one line $(x^k=a\cdot x^l)$. In what I need we can assume that all points lie on a sphere.
Specifically, my motivation is that looking at eigenspaces of adjacency matrix, we can convert the graph isomorphism problem into a question if two sets of points differ only by rotation (page 9-11 here. For strongly regular graphs these points are on a sphere and form a very regular polyhedron. Hence, I wanted to use an affine space of quadratic forms defining "wobbling" ellipsoids, such that they intersect only in our set - then we could use characteristic polynomial to test if they differ only by rotation. The crucial question is if e.g. $\{x: x^T A x=1 \textrm{ for all }A=A_0 + a\cdot A_1\}$ doesn't add too many extra points to the description. Geometrically: if "wobbling" ellipsoids with fixed some points, what extra fixed points would their intersection have?
Here is example of 2D situation: describing two points as intersection of ellipses/hyperbolas. Intersection only adds symmetric ($-x$) points, the question is when it is true, also in higher dimensions: