## Enumeration of algebraic and tropical singular hypersurfaces

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- by Uriel Sinichkin PDF
- Trans. Amer. Math. Soc.
**375**(2022), 8529-8580

## Abstract:

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.## References

- Alicia Dickenstein and Luis F. Tabera,
*Singular tropical hypersurfaces*, Discrete Comput. Geom.**47**(2012), no. 2, 430–453. MR**2872547**, DOI 10.1007/s00454-011-9364-6 - I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky,
*Discriminants, resultants and multidimensional determinants*, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008. Reprint of the 1994 edition. MR**2394437** - Branko Grünbaum,
*Convex polytopes*, 2nd ed., Graduate Texts in Mathematics, vol. 221, Springer-Verlag, New York, 2003. Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler. MR**1976856**, DOI 10.1007/978-1-4613-0019-9 - Ó. Iglesias Valiño and F. Santos,
*The complete classification of empty lattice 4-simplices*, Electron. Notes Discrete Math.**68**(2018), 155–160. - Ilia Itenberg, Grigory Mikhalkin, and Eugenii Shustin,
*Tropical algebraic geometry*, 2nd ed., Oberwolfach Seminars, vol. 35, Birkhäuser Verlag, Basel, 2009. MR**2508011**, DOI 10.1007/978-3-0346-0048-4 - Christian U. Jensen and Helmut Lenzing,
*Model-theoretic algebra with particular emphasis on fields, rings, modules*, Algebra, Logic and Applications, vol. 2, Gordon and Breach Science Publishers, New York, 1989. MR**1057608** - D. Kerner, Personal communication, 2017.
- Diane Maclagan and Bernd Sturmfels,
*Introduction to tropical geometry*, Graduate Studies in Mathematics, vol. 161, American Mathematical Society, Providence, RI, 2015. MR**3287221**, DOI 10.1090/gsm/161 - H. Markwig, T. Markwig, and E. Shustin,
*Tropical curves with a singularity in a fixed point*, Manuscripta Math.**137**(2012), no. 3, 383–418. - Hannah Markwig, Thomas Markwig, and Eugenii Shustin,
*Tropical curves with a singularity in a fixed point*, Manuscripta Math.**137**(2012), no. 3-4, 383–418. MR**2875284**, DOI 10.1007/s00229-011-0471-8 - Hannah Markwig, Thomas Markwig, and Eugenii Shustin,
*Tropical surface singularities*, Discrete Comput. Geom.**48**(2012), no. 4, 879–914. MR**3000569**, DOI 10.1007/s00454-012-9453-1 - Hannah Markwig, Thomas Markwig, and Eugenii Shustin,
*Enumeration of complex and real surfaces via tropical geometry*, Adv. Geom.**18**(2018), no. 1, 69–100. MR**3750255**, DOI 10.1515/advgeom-2017-0024 - Grigory Mikhalkin,
*Counting curves via lattice paths in polygons*, C. R. Math. Acad. Sci. Paris**336**(2003), no. 8, 629–634 (English, with English and French summaries). MR**1988122**, DOI 10.1016/S1631-073X(03)00104-3 - Grigory Mikhalkin,
*Enumerative tropical algebraic geometry in $\Bbb R^2$*, J. Amer. Math. Soc.**18**(2005), no. 2, 313–377. MR**2137980**, DOI 10.1090/S0894-0347-05-00477-7 - E. Shustin,
*A tropical approach to enumerative geometry*, Algebra i Analiz**17**(2005), no. 2, 170–214; English transl., St. Petersburg Math. J.**17**(2006), no. 2, 343–375. MR**2159589**, DOI 10.1090/S1061-0022-06-00908-3 - O. Viro,
*Patchworking Real Algebraic Varieties*, Appendix to G.-M. Greuel et al. Singular Algebraic Curves, Springer, Switzerland, 2018. - G. M. Ziegler,
*Lectures On Polytopes*, Grad. Texts in Math., Springer, 2007.

## Additional Information

**Uriel Sinichkin**- Affiliation: School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
- ORCID: 0000-0001-6892-5665
- Email: sinichkin@mail.tau.ac.il
- 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: https://doi.org/10.1090/tran/8753