Probabilistic enumerative geometry over p-adic numbers: linear spaces on complete intersections
Annales Henri Lebesgue, Volume 5 (2022), pp. 1329-1360.



We compute the expectation of the number of linear spaces on a random complete intersection in p-adic projective space. Here “random” means that the coefficients of the polynomials defining the complete intersections are sampled uniformly from the p-adic integers. We show that as the prime p tends to infinity the expected number of linear spaces on a random complete intersection tends to 1. In the case of the number of lines on a random cubic in three-space and on the intersection of two random quadrics in four-space, we give an explicit formula for this expectation.


[AEMBM21] Ait El Manssour, Rida; Belotti, Mara; Meroni, Chiara Real Lines on Random Cubic Surfaces, Arnold Math J., Volume 7 (2021) no. 4, pp. 541-559 | DOI | MR | Zbl

[BD20] Brandes, Julia; Dietmann, Rainer Rational lines on cubic hypersurfaces, Math. Proc. Camb. Philos. Soc. (2020), p. 1–14 | DOI | Zbl

[BL20] Bürgisser, Peter; Lerario, Antonio Probabilistic Schubert calculus, J. Reine Angew. Math., Volume 760 (2020), pp. 1-58 | DOI | MR | Zbl

[BLLP19] Basu, Saugata; Lerario, Antonio; Lundberg, Erik; Peterson, Chris Random fields and the enumerative geometry of lines on real and complex hypersurfaces, Math. Ann., Volume 374 (2019) no. 3-4, pp. 1773-1810 | DOI | MR | Zbl

[Car22] Caruso, Xavier Where are the zeroes of a random p-adic polynomial?, Forum Math. Sigma, Volume 10 (2022), e55 | DOI | MR | Zbl

[DM98] Debarre, Olivier; Manivel, Laurent Sur la variété des espaces linéaires contenus dans une intersection complète, Math. Ann., Volume 312 (1998) no. 3, pp. 549-574 | DOI | MR | Zbl

[EK95] Edelman, Alan; Kostlan, Eric How many zeros of a random polynomial are real?, Bull. Am. Math. Soc., Volume 32 (1995) no. 1, pp. 1-37 | DOI | MR | Zbl

[EKS94] Edelman, Alan; Kostlan, Eric; Shub, Michael How many eigenvalues of a random matrix are real?, J. Am. Math. Soc., Volume 7 (1994) no. 1, pp. 247-267 | DOI | MR | Zbl

[EMT19] El Maazouz, Yassine; Tran, Ngoc Mai Statistics of Gaussians on local fields and their tropicalizations (2019) (

[Eva02] Evans, Steven N. Elementary divisors and determinants of random matrices over a local field, Stochastic Processes Appl., Volume 102 (2002) no. 1, p. 89-02 | DOI | MR | Zbl

[Eva06] Evans, Steven N. The expected number of zeros of a random system of p-adic polynomials, Electron. Commun. Probab., Volume 11 (2006), pp. 278-290 | DOI | MR | Zbl

[FK13] Finashin, Sergey; Kharlamov, Viatcheslav Abundance of real lines on real projective hypersurfaces, Int. Math. Res. Not. (2013) no. 16, pp. 3639-3646 | DOI | MR | Zbl

[Kac43] Kac, Mark On the average number of real roots of a random algebraic equation, Bull. Am. Math. Soc., Volume 49 (1943), pp. 314-320 | DOI | MR | Zbl

[KL21] Kulkarni, Avinash; Lerario, Antonio p-adic Integral Geometry, SIAM J. Appl. Algebra Geom., Volume 5 (2021) no. 1, pp. 28-59 | DOI | MR | Zbl

[Kos93] Kostlan, Eric On the distribution of roots of random polynomials, From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990), Springer, 1993, pp. 419-431 | DOI | MR | Zbl

[KW21] Kass, Jesse Leo; Wickelgren, Kirsten An Arithmetic Count of the Lines on a Smooth Cubic Surface, Compos. Math., Volume 157 (2021) no. 4, pp. 677-709 | DOI | MR | Zbl

[OT14] Okonek, Christian; Teleman, Andrei Intrinsic signs and lower bounds in real algebraic geometry, J. Reine Angew. Math., Volume 688 (2014), pp. 219-241 | DOI | MR | Zbl

[Seg42] Segre, Benjamino The Non-singular Cubic Surfaces, Oxford University Press, 1942 | MR | Zbl

[SS93] Shub, Michael; Smale, Steve Complexity of Bezout’s theorem. III. Condition number and packing, J. Complexity, Volume 9 (1993) no. 1, pp. 4-14 (Festschrift for Joseph F. Traub, Part I) | DOI | MR | Zbl