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.References
- David Cox, John Little, and Donal O’Shea, Using algebraic geometry, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, New York, 1998. MR 1639811, DOI 10.1007/978-1-4757-6911-1
- Edward D. Davis and Anthony V. Geramita, Birational morphisms to $\textbf {P}^2$: an ideal-theoretic perspective, Math. Ann. 279 (1988), no. 3, 435–448. MR 922427, DOI 10.1007/BF01456280
- Herbert Edelsbrunner, Algorithms in combinatorial geometry, EATCS Monographs on Theoretical Computer Science, vol. 10, Springer-Verlag, Berlin, 1987. MR 904271, DOI 10.1007/978-3-642-61568-9
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
- David Eisenbud, The geometry of syzygies, Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York, 2005. A second course in commutative algebra and algebraic geometry. MR 2103875
- Leah Gold, John Little, and Hal Schenck, Cayley-Bacharach and evaluation codes on complete intersections, J. Pure Appl. Algebra 196 (2005), no. 1, 91–99. MR 2111849, DOI 10.1016/j.jpaa.2004.08.015
- D. R. Grayson and M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/.
- Leonidas J. Guibas, Mark H. Overmars, and Jean-Marc Robert, The exact fitting problem in higher dimensions, Comput. Geom. 6 (1996), no. 4, 215–230. MR 1392311, DOI 10.1016/0925-7721(95)00020-8
- Johan P. Hansen, Linkage and codes on complete intersections, Appl. Algebra Engrg. Comm. Comput. 14 (2003), no. 3, 175–185. MR 2013791, DOI 10.1007/s00200-003-0119-3
- Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR 1217488, DOI 10.1007/978-3-662-02772-1
- Hal Schenck, Resonance varieties via blowups of $\Bbb P^2$ and scrolls, Int. Math. Res. Not. IMRN 20 (2011), 4756–4778. MR 2844937, DOI 10.1093/imrn/rnq271
- Ştefan O. Tohǎneanu, Lower bounds on minimal distance of evaluation codes, Appl. Algebra Engrg. Comm. Comput. 20 (2009), no. 5-6, 351–360. MR 2564409, DOI 10.1007/s00200-009-0102-8
- Ştefan O. Tohǎneanu, On the de Boer-Pellikaan method for computing minimum distance, J. Symbolic Comput. 45 (2010), no. 10, 965–974. MR 2679386, DOI 10.1016/j.jsc.2010.06.021
- Ştefan O. Tohǎneanu, The minimum distance of sets of points and the minimum socle degree, J. Pure Appl. Algebra 215 (2011), no. 11, 2645–2651. MR 2802154, DOI 10.1016/j.jpaa.2011.03.008
- Ştefan O. Tohǎneanu and Adam Van Tuyl, Bounding invariants of fat points using a coding theory construction, J. Pure Appl. Algebra 217 (2013), no. 2, 269–279. MR 2969252, DOI 10.1016/j.jpaa.2012.06.004
- Wolmer V. Vasconcelos, Computational methods in commutative algebra and algebraic geometry, Algorithms and Computation in Mathematics, vol. 2, Springer-Verlag, Berlin, 1998. With chapters by David Eisenbud, Daniel R. Grayson, Jürgen Herzog and Michael Stillman. MR 1484973, DOI 10.1007/978-3-642-58951-5
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
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3134006