A commutative algebraic approach to the fitting problem

Tohǎneanu, Ştefan

doi:10.1090/S0002-9939-2013-11814-9

A commutative algebraic approach to the fitting problem
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by Ştefan O. Tohǎneanu PDF

Proc. Amer. Math. Soc. 142 (2014), 659-666 Request permission

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.

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