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A commutative algebraic approach to the fitting problem


Author: Ştefan O. Tohǎneanu
Journal: Proc. Amer. Math. Soc. 142 (2014), 659-666
MSC (2010): Primary 52C35; Secondary 13P25, 13P20
DOI: https://doi.org/10.1090/S0002-9939-2013-11814-9
Published electronically: October 25, 2013
MathSciNet review: 3134006
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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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Additional Information

Ştefan O. Tohǎneanu
Affiliation: Department of Mathematics, The University of Western Ontario, London, ON N6A 5B7, Canada
Address at time of publication: Department of Mathematics, University of Idaho, 875 Perimeter Drive, MS1103, Moscow, Idaho 83844-1103
Email: stohanea@uwo.ca, tohaneanu@uidaho.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11814-9
Keywords: Index of nilpotency, fat points, minimum distance
Received by editor(s): March 1, 2012
Received by editor(s) in revised form: March 22, 2012
Published electronically: October 25, 2013
Communicated by: Irena Peeva
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.