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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Enumeration of algebraic and tropical singular hypersurfaces
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by Uriel Sinichkin PDF
Trans. Amer. Math. Soc. 375 (2022), 8529-8580


We develop a version of Mikhalkin’s lattice path algorithm for projective hypersurfaces of arbitrary degree and dimension, which enumerates singular tropical hypersurfaces passing through appropriate configuration of points. By proving a correspondence theorem combined with the lattice path algorithm, we construct a $\delta$ dimensional linear space of degree $d$ real hypersurfaces containing $\frac {1}{\delta !}(\gamma _nd^n)^{\delta }+O(d^{n\delta -1})$ hypersurfaces with $\delta$ real nodes, where $\gamma _n$ are positive and given by a recursive formula. This is asymptotically comparable to the number $\frac {1}{\delta !} \left ( (n+1)(d-1)^n \right )^{\delta }+O\left (d^{n(\delta -1)} \right )$ of complex hypersurfaces having $\delta$ nodes in a $\delta$ dimensional linear space. In the case $\delta =1$ we give a slightly better leading term.
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Additional Information
  • Uriel Sinichkin
  • Affiliation: School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
  • ORCID: 0000-0001-6892-5665
  • Email:
  • Received by editor(s): November 21, 2020
  • Received by editor(s) in revised form: January 26, 2022
  • Published electronically: September 23, 2022
  • Additional Notes: The research was supported by Israel Science Foundation grant number 501/18 and by the Bauer-Neuman Chair in Real and Complex Geometry.
  • © Copyright 2022 by the authors
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 8529-8580
  • MSC (2020): Primary 14N10; Secondary 14P05, 14T90, 14J17, 05E14, 52B20
  • DOI: