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.


  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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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.


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Information & Authors


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


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


  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



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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