Abstract

Motivated by applications in combinatorial geometry, we consider the following question: Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m)$ be an $m$-partition of a positive integer $n$, $S_i \subseteq \mathbb{C}^{\lambda_i}$ be finite sets, and let $S:=S_1 \times S_2 \times \cdots \times S_m \subset \mathbb{C}^n$ be the multigrid defined by $S_i$. Suppose $p$ is an $n$-variate degree $d$ polynomial. How many zeros does $p$ have on $S$? We first develop a multivariate generalization of the combinatorial nullstellensatz that certifies existence of a point $t \in S$ so that $p(t) \neq 0$. Then we show that a natural multivariate generalization of the DeMillo--Lipton--Schwartz--Zippel lemma holds, except for a special family of polynomials that we call $\lambda$-reducible. This yields a simultaneous generalization of the Szemerédi--Trotter theorem and the Schwartz--Zippel lemma into higher dimensions, and has applications in incidence geometry. Finally, we develop a symbolic algorithm that identifies certain $\lambda$-reducible polynomials. More precisely, our symbolic algorithm detects polynomials that include a Cartesian product of hypersurfaces in their zero set. It is likely that using Chow forms the algorithm can be generalized to handle arbitrary $\lambda$-reducible polynomials, which we leave as an open problem.

Keywords

  1. Schwartz--Zippel lemma
  2. combinatorial nullstellensatz
  3. combinatorial geometry
  4. polynomial partitioning
  5. incidence geometry
  6. resultant
  7. generalized characteristic polynomial

MSC codes

  1. 68Q25
  2. 68R10
  3. 68U05

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References

1.
N. Alon, Combinatorial Nullstellensatz, Combin. Probab. Comput., 8 (1999), pp. 7--29.
2.
S. Barone and S. Basu, Refined bounds on the number of connected components of sign conditions on a variety, Discrete Comput. Geom., 47 (2012), pp. 577--597.
3.
J. Canny, Generalised characteristic polynomials, J. Symbolic Comput., 9 (1990), pp. 241--250.
4.
Z. Dvir, On the size of Kakeya sets in finite fields, J. Amer. Math. Soc., 22 (2009), pp. 1093--1097.
5.
Z. Dvir, Incidence theorems and their applications, Found. Trends. Theor. Comput. Sci., 6 (2012), pp. 257--393.
6.
D. Eisenbud, C. Huneke, and W. Vasconcelos, Direct methods for primary decomposition, Invent. Math., 110 (1992), pp. 207--235.
7.
I. Z. Emiris and B. Mourrain, Matrices in elimination theory, J. Symbolic Comput., 28 (1999), pp. 3--44.
8.
J. Fox, J. Pach, A. Sheffer, A. Suk, and J. Zahl, A semi-algebraic version of Zarankiewicz's problem, J. Eur. Math. Soc. (JEHS), 19 (2017), pp. 1785--1810.
9.
I. M. Gelfand, M. Kapranov, and A. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Springer Boston, 2008.
10.
L. Guth, Polynomial Methods in Combinatorics, Univ. Lecture Ser. 64, American Mathematical Soc., Providence, RI, 2016.
11.
L. Guth and N. H. Katz, Algebraic methods in discrete analogs of the Kakeya problem, Adv. Math. (2), 225 (2010), pp. 2828--2839.
12.
L. Guth and N. H. Katz, On the Erdös distinct distances problem in the plane, Ann. Math., 181 (2015), pp. 155--190.
13.
14.
H. N. Mojarrad, T. Pham, C. Valculescu, and F. de Zeeuw, Schwartz--Zippel bounds for two-dimensional products, Discrete Anal., (2017).
15.
D. Mumford, Algebraic Geometry I: Complex Projective Varieties, Springer Science & Business Media, 1995. Reprint of the 1976 edition.
16.
J. Pach and M. Sharir, On the number of incidences between points and curves, Combin., Probab. Comput., 7 (1998), pp. 121--127.
17.
O. E. Raz, M. Sharir, and J. Solymosi, Polynomials vanishing on grids: The Elekes-Rónyai problem revisited, in Proceedings of the Thirtieth Annual Symposium on Computational Geometry, ACM, New York, 2014, pp. 160--251.
18.
M.-F. Roy and N. Vorobjov, The complexification and degree of a semi-algebraic set, Math. Zeit., 239 (2002), pp. 131--142.
19.
N. Saxena, Progress on polynomial identity testing., Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 99 (2009), pp. 49--79.
20.
J. Schmid, On the affine Bézout inequality, Manuscripta Math., 88 (1995), pp. 225--232.
21.
A. Sheffer, Polynomial Methods and Incidence Theory, to appear, 2019, http://faculty.baruch.cuny.edu/ASheffer/000book.pdf.
22.
A. Sheffer, E. Szabó, and J. Zahl, Point-curve incidences in the complex plane, Combinatorica, 38 (2018), pp. 487--499.
23.
J. Solymosi and T. Tao, An incidence theorem in higher dimensions, Discrete Comput. Geom., 48 (2012), pp. 255--280.
24.
J. Spencer, E. Szemerédi, and W. T. Trotter, Unit distances in the Euclidean plane, in Graph Theory and Combinatorics, Academic Press, London, 1984, pp. 293--303.
25.
E. Szemerédi and W. T. Trotter, Extremal problems in discrete geometry, Combinatorica, 3 (1983), pp. 381--392.
27.
T. Tao, Algebraic combinatorial geometry: The polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory, EMS Surv. Math. Sci., 1 (2014), pp. 1--46.
28.
C. D. Tóth, The Szemerédi-Trotter theorem in the complex plane, Combinatorica, 35 (2015), pp. 95--126.
29.
W. Vogel and D. P. Patil, Remarks on the algebraic approach to intersection theory., Monatsh. Math., 96 (1983), pp. 233--250, http://eudml.org/doc/178150.

Information & Authors

Information

Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 888 - 910
ISSN (online): 1095-7146

History

Submitted: 24 April 2020
Accepted: 8 November 2021
Published online: 7 April 2022

Keywords

  1. Schwartz--Zippel lemma
  2. combinatorial nullstellensatz
  3. combinatorial geometry
  4. polynomial partitioning
  5. incidence geometry
  6. resultant
  7. generalized characteristic polynomial

MSC codes

  1. 68Q25
  2. 68R10
  3. 68U05

Authors

Affiliations

Funding Information

PGMO : ALMA
PHC GRAPE
Agence Nationale de la Recherche https://doi.org/10.13039/501100001665 : ANR-17-CE40-0009
Horizon 2020 Framework Programme https://doi.org/10.13039/100010661 : 787840
Einstein Stiftung Berlin https://doi.org/10.13039/501100006188
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1460766

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