It is obvious that $d+1$ points uniquely determine a circumscribed hypersphere in $d$ -dimensional space. If the number of points $n \le d$, the circumscribed hypersphere is not unique, but the smallest one exists and must have its center lying inside the space spanned by these $n$ points.
For example, when the hypersphere is unique and $n=4, d=3$ (i.e. $n=d+1$), and these four points are denoted as $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$, respectively, it is clear that the unique circumscribed equation is given by $\textrm{InSphere}(\textbf{e})=0$ where $\textrm{InSphere}(\textbf{e})$ is
As we can see, the hypersphere equation is given by a determinant for cases where $n=d+1$. We can expect the hypersphere where $n\le d$ might be similarly derived and it should be a simple determinant equation as well.
Do you have any ideas on how to derive such a hypersphere (the smallest one) in the form of determinant equation where $n\le d$?
Isn't this is a trivial problem? If you have $n$ points with no $3$ on a line, $4$ on a plane, etc... then they lie on a $n-2$ sphere, and the smallest spheres in higher dimensions will be those with the same center and radius as this $n-2$ sphere.