A commutative algebraic approach to the fitting problem

TL;DR

This paper introduces an algebraic method using ideals and nilpotency to determine the maximum number of points from a finite set in projective space that can lie on a hyperplane, providing a new algebraic perspective on the fitting problem.

Contribution

It establishes a novel algebraic framework linking the fitting problem to the nilpotency index of an ideal derived from point coordinates.

Findings

Maximum collinear points in P^2 equal nilpotency index plus one.

For sets with points spanning hyperplanes, maximum points on a hyperplane relate to ideal nilpotency.

Provides algebraic formulas for the fitting problem in projective spaces.

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.