Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Published electronically: October 25, 2013
MathSciNet review: 3134006
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] David Cox, John Little, and Donal O'Shea, Using algebraic geometry, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, New York, 1998. MR 1639811 (99h:13033)
  • [2] Edward D. Davis and Anthony V. Geramita, Birational morphisms to $ {\bf P}^2$: an ideal-theoretic perspective, Math. Ann. 279 (1988), no. 3, 435-448. MR 922427 (89a:14014),
  • [3] Herbert Edelsbrunner, Algorithms in combinatorial geometry, EATCS Monographs on Theoretical Computer Science, vol. 10, Springer-Verlag, Berlin, 1987. MR 904271 (89a:68205)
  • [4] David Eisenbud, Commutative algebra, With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. MR 1322960 (97a:13001)
  • [5] David Eisenbud, The geometry of syzygies, A second course in commutative algebra and algebraic geometry. Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York, 2005. MR 2103875 (2005h:13021)
  • [6] 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 (2005k:14050),
  • [7] D. R. Grayson and M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry,
  • [8] 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 (98c:52020),
  • [9] Johan P. Hansen, Linkage and codes on complete intersections, Appl. Algebra Engrg. Comm. Comput. 14 (2003), no. 3, 175-185. MR 2013791 (2004j:13018),
  • [10] 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 (94e:52014)
  • [11] Hal Schenck, Resonance varieties via blowups of $ \mathbb{P}^2$ and scrolls, Int. Math. Res. Not. IMRN 20 (2011), 4756-4778. MR 2844937 (2012k:14075)
  • [12] Ştefan O. Tohaneanu, Lower bounds on minimal distance of evaluation codes, Appl. Algebra Engrg. Comm. Comput. 20 (2009), no. 5-6, 351-360. MR 2564409 (2011c:13027),
  • [13] Ştefan O. Tohaneanu, On the de Boer-Pellikaan method for computing minimum distance, J. Symbolic Comput. 45 (2010), no. 10, 965-974. MR 2679386 (2011g:94074),
  • [14] Ştefan O. Tohaneanu, The minimum distance of sets of points and the minimum socle degree, J. Pure Appl. Algebra 215 (2011), no. 11, 2645-2651. MR 2802154 (2012e:13028),
  • [15] Ştefan O. Tohaňeanu 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,
  • [16] 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 (99c:13048)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 52C35, 13P25, 13P20

Retrieve articles in all journals with MSC (2010): 52C35, 13P25, 13P20

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

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.

American Mathematical Society