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A commutative algebraic approach to the fitting problem

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