Abstract.

We describe a framework for estimating Hilbert–Samuel multiplicities \(\mathbf{e}_XY\) for pairs of projective varieties \(X \subset Y\) from finite point samples rather than defining equations. The first step involves proving that this multiplicity remains invariant under certain hyperplane sections which reduce \(X\) to a point \(p\) and \(Y\) to a curve \(C\) . Next, we establish that \(\mathbf{e}_pC\) equals the Euler characteristic (and hence the cardinality) of the complex link of \(p\) in \(C\) . Finally, we provide explicit bounds on the number of uniform point samples needed (in an annular neighborhood of \(p\) in \(C\) ) to determine this Euler characteristic with high confidence.

Keywords

  1. intersection multiplicity
  2. stratified Morse theory
  3. complex links
  4. Hilbert–Samuel multiplicity

MSC codes

  1. 14B05
  2. 57N80
  3. 13H15
  4. 32S60
  5. 14J17

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Acknowledgments.

Mark Goresky kindly shared an advance copy of his survey Morse theory, stratifications and sheaves [8] with us; that paper served as our Polaris while we navigated the formidable waters surrounding these topics. We are grateful to Heather Harrington, Kate Turner, and Yossi Bokor for facilitating this work in its early stages. We also thank the Sydney Mathematics Research Institute (SMRI) at the University of Sydney for their generous hospitality. This paper benefited enormously from the suggestions and corrections provided by the two anonymous referees.

References

1.
P. Breiding, S. Kališnik, B. Sturmfels, and M. Weinstein, Learning algebraic varieties from samples, Rev. Mat. Complut., 31 (2018), pp. 545–593.
2.
D. A. Cox, J. Little, and D. O’Shea, Using Algebraic Geometry, Grad. Texts Math. 185, Springer, New York, 2005.
3.
E. Dufresne, P. Edwards, H. Harrington, and J. Hauenstein, Sampling real algebraic varieties for topological data analysis, in Proceedings of the 18th IEEE International Conference on Machine Learning and Applications (ICMLA), 2019, pp. 1531–1536.
4.
D. Eisenbud and J. Harris, 3264 and All That: A Second Course in Algebraic Geometry, Cambridge University Press, Cambridge, UK, 2016.
5.
H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), pp. 418–491.
6.
W. Fulton, Introduction to Intersection Theory in Algebraic Geometry, CBMS Regional Conf. Ser. in Math. 54, American Mathematical Society, Providence, RI, 1984.
7.
W. Fulton, Intersection Theory, 2nd ed., Springer-Verlag, Berlin, 1998.
8.
M. Goresky, Morse theory, stratifications and sheaves, in Handbook of Geometry and Topology of Singularities, J. L. Cisneros-Molina, L. D. Tráng, and J. Seade, eds., Springer, Cham, 2020, pp. 275–319.
9.
M. Goresky and R. MacPherson, Stratified Morse Theory, Springer-Verlag, Berlin, 1988.
10.
C. Harris, Computing Segre classes in arbitrary projective varieties, J. Symbolic Comput., 82 (2017), pp. 26–37.
11.
C. Harris and M. Helmer, Segre class computation and practical applications, Math. Comp., 89 (2020), pp. 465–491.
12.
R. Hartshorne, Algebraic Geometry, Grad. Texts. Math. 52, Springer, New York, 2013.
13.
M. Helmer and V. Nanda, Conormal spaces and Whitney stratifications, Found. Comput. Math., (2022), https://doi.org/10.1007/s10208-022-09574-8.
14.
M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren Math. Wiss. 292, Springer-Verlag, Berlin, 1990.
15.
D. Massey, Characteristic cycles and the relative local Euler obstruction, in A Panorama of Singularities: Conference in Celebration of Lê Dũng Tráng’s 70th Birthday, Universidad de Sevilla, Sevilla, Spain, 2020, pp. 137–156.
16.
J. Mather, Notes on topological stability, Bull. Amer. Math. Soc., 49 (2012), pp. 475–506.
17.
G. Moroz, F. Rouiller, D. Chablat, and P. Wenger, On the determination of cusp points of 3-RPR parallel manipulators, Mech. Mach. Theory, 45 (2010), pp. 1555–1567.
18.
P. Niyogi, S. Smale, and S. Weinberger, Finding the homology of submanifolds with high confidence from random samples, Discrete Comput. Geom., 39 (2008), pp. 419–441.
19.
S. Piipponen, T. Arponen, and J. Tuomela, Classification of singularities in kinematics of mechanisms, in Computational Kinematics, Springer, Dordrecht, 2014, pp. 41–48.
20.
F. Rouillier, Solving zero-dimensional systems through the rational univariate representation, Appl. Algebra Eng. Commun. Comput., 9 (1999), pp. 433–461.
21.
P. Samuel, Méthodes d’algèbre abstraite en géométrie algébrique, Ergebnisse der Mathematik, Springer-Verlag, Berlin, Heidelberg, 1955.
22.
J.-P. Serre, Local Algebra, Springer-Verlag, Berlin, 2012.
23.
I. R. Shafarevich, Basic Algebraic Geometry, Vol. 2, Springer-Verlag, Berlin, 1994.
24.
A. J. Sommese and C. W. Wampler, The Numerical Solution of Systems of Polynomials Arising in Engineering and Science, World Scientific, Hackensack, NJ, 2005.
25.
B. J. Stolz, J. Tanner, H. A. Harrington, and V. Nanda, Geometric anomaly detection in data, Proc. Natl. Acad. Sci. USA, 117 (2020), pp. 19664–19669.
26.
C. W. Wampler and A. J. Sommese, Numerical algebraic geometry and algebraic kinematics, Acta Numer., 20 (2011), pp. 469–567.
27.
Y. Wang and B. Wang, Topological inference of manifolds with boundary, Comput. Geom., 88 (2020), 101606.
28.
H. Whitney, Tangents to an analytic variety, Ann. of Math. (2), 81 (1965), pp. 496–549.
29.
M. Zein, P. Wenger, and D. Chablat, Singular curves in the joint space and cusp points of 3-RPR parallel manipulators, Robotica, 25 (2007), pp. 717–724.

Information & Authors

Information

Published In

cover image SIAM Journal on Applied Algebra and Geometry
SIAM Journal on Applied Algebra and Geometry
Pages: 29 - 48
ISSN (online): 2470-6566

History

Submitted: 2 February 2022
Accepted: 10 November 2022
Published online: 8 March 2023

Keywords

  1. intersection multiplicity
  2. stratified Morse theory
  3. complex links
  4. Hilbert–Samuel multiplicity

MSC codes

  1. 14B05
  2. 57N80
  3. 13H15
  4. 32S60
  5. 14J17

Authors

Affiliations

Martin Helmer
Department of Mathematics, North Carolina State University, Raleigh, NC 27695 USA.
Vidit Nanda Contact the author
Mathematical Institute, University of Oxford, Oxford, UK, OX2 6GG.

Funding Information

EPSRC: EP/R018472/1
Funding: The work of the second author was supported by EPSRC grant EP/R018472/1 and by DSTL grant D015 funded through the Alan Turing Institute.

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