A commutative algebraic approach to the fitting problem

Abstract

Given a finite set of points $\Gamma$ in $\mathbb P^{k-1}$ not all contained in a hyperplane, the "fitting problem" asks what is the maximum number $hyp(\Gamma)$ of these points that can fit in some hyperplane and what is (are) the equation(s) of such hyperplane(s). If $\Gamma$ has the property that any $k-1$ of its points span a hyperplane, then $hyp(\Gamma)=nil(I)+k-2$, where $nil(I)$ is the index of nilpotency of an ideal constructed from the homogeneous coordinates of the points of $\Gamma$. Note that in $\mathbb P^2$ any two points span a line, and we find that the maximum number of collinear points of any given set of points $\Gamma\subset\mathbb P^2$ equals the index of nilpotency of the corresponding ideal, plus one.