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# Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets

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## Acknowledgements

The authors wish to thank Adam Sheffer and Frank de Zeeuw for suggesting the transformation used for collinearity testing in Theorem 6.1. We also thank Jean Cardinal, John Iacono, Stefan Langerman, and Aurélien Ooms for useful discussions.

## Cite As

Boris Aronov, Esther Ezra, and Micha Sharir. Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.8

## Abstract

We present subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets A, B, and C of points in the plane, that satisfy a certain polynomial equation or two equations. The best known instance of such a problem is testing for the existence of a collinear triple of points in A×B×C, a classical 3SUM-hard problem that has so far defied any attempt to obtain a subquadratic solution, whether in the (uniform) real RAM model, or in the algebraic decision-tree model. While we are still unable to solve this problem, in full generality, in subquadratic time, we obtain such a solution, in the algebraic decision-tree model, that uses only roughly O(n^(28/15)) constant-degree polynomial sign tests, for the special case where two of the sets lie on one-dimensional curves and the third is placed arbitrarily in the plane. Our technique is fairly general, and applies to any other problem where we seek a triple that satisfies a single polynomial equation, e.g., determining whether A× B× C contains a triple spanning a unit-area triangle. This result extends recent work by Barba et al. [Luis Barba et al., 2019] and by Chan [Timothy M. Chan, 2020], where all three sets A, B, and C are assumed to be one-dimensional. While there are common features in the high-level approaches, here and in [Luis Barba et al., 2019], the actual analysis in this work becomes more involved and requires new methods and techniques, involving polynomial partitions and other related tools. As a second application of our technique, we again have three n-point sets A, B, and C in the plane, and we want to determine whether there exists a triple (a,b,c) ∈ A×B×C that simultaneously satisfies two real polynomial equations. For example, this is the setup when testing for the existence of pairs of similar triangles spanned by the input points, in various contexts discussed later in the paper. We show that problems of this kind can be solved with roughly O(n^(24/13)) constant-degree polynomial sign tests. These problems can be extended to higher dimensions in various ways, and we present subquadratic solutions to some of these extensions, in the algebraic decision-tree model.

## Subject Classification

##### ACM Subject Classification
• Theory of computation
• Theory of computation → Computational geometry
##### Keywords
• Algebraic decision tree
• Polynomial partition
• Collinearity testing
• 3SUM-hard problems
• Polynomials vanishing on Cartesian products

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## References

1. Pankaj K. Agarwal, Jiří Matoušek, and Micha Sharir. On range searching with semialgebraic sets. II. SIAM J. Comput., 42(6):2039-2062, 2013. URL: https://doi.org/10.1137/120890855.
2. Boris Aronov and Jean Cardinal. Geometric pattern matching reduces to k-SUM, 2020. URL: http://arxiv.org/abs/abs/2003.11890.
3. Boris Aronov, Esther Ezra, and Micha Sharir. Testing polynomials for vanishing on Cartesian products of planar point sets, 2020. Full version of this paper. URL: http://arxiv.org/abs/2003.09533.
4. Luis Barba, Jean Cardinal, John Iacono, Stefan Langerman, Aurélien Ooms, and Noam Solomon. Subquadratic algorithms for algebraic 3SUM. Discrete Comput. Geom., 61(4):698-734, 2019. URL: https://doi.org/10.1007/s00454-018-0040-y.
5. Saugata Basu, Richard Pollack, and Marie-Françoise Roy. Algorithms in Real Algebraic Geometry, volume 10 of Algorithms and Computation in Mathematics. Springer-Verlag, Berlin, second edition, 2006.
6. Michael Ben-Or. Lower bounds for algebraic computation trees (preliminary report). In Proc. of the 15th Annual ACM Symposium on Theory of Computing, pages 80-86, 1983. URL: https://doi.org/10.1145/800061.808735.
7. Timothy M. Chan. More logarithmic-factor speedups for 3SUM, (median, +)-convolution, and some geometric 3SUM-hard problems. ACM Trans. Algorithms, 16(1):7:1-7:23, 2020. URL: https://doi.org/10.1145/3363541.
8. Bernard Chazelle. Cutting hyperplanes for divide-and-conquer. Discrete Comput. Geom., 9:145-158, 1993. URL: https://doi.org/10.1007/BF02189314.
9. Frank de Zeeuw. Ordinary lines in space, 2018. URL: http://arxiv.org/abs/1803.09524.
10. György Elekes and Lajos Rónyai. A combinatorial problem on polynomials and rational functions. J. Comb. Theory, Ser. A, 89(1):1-20, 2000. URL: https://doi.org/10.1006/jcta.1999.2976.
11. György Elekes and Endre Szabó. How to find groups? (and how to use them in Erdős geometry?). Combinatorica, 32(5):537-571, 2012. URL: https://doi.org/10.1007/s00493-012-2505-6.
12. Jeff Erickson and Raimund Seidel. Better lower bounds on detecting affine and spherical degeneracies. Discrete Comput. Geom., 13:41-57, 1995. URL: https://doi.org/10.1007/BF02574027.
13. Ari Freund. Improved subquadratic 3SUM. Algorithmica, 77(2):440-458, 2017. URL: https://doi.org/10.1007/s00453-015-0079-6.
14. Anka Gajentaan and Mark H. Overmars. On a class of O(n²) problems in computational geometry. Comput. Geom., 5:165-185, 1995. URL: https://doi.org/10.1016/0925-7721(95)00022-2.
15. Omer Gold and Micha Sharir. Improved bounds for 3SUM, k-SUM, and linear degeneracy. In 25th Annual European Symposium on Algorithms, pages 42:1-42:13, 2017. Also in arXiv:1512.05279. URL: https://doi.org/10.4230/LIPIcs.ESA.2017.42.
16. Allan Grønlund and Seth Pettie. Threesomes, degenerates, and love triangles. J. ACM, 65(4):22:1-22:25, 2018. URL: https://doi.org/10.1145/3185378.
17. Larry Guth and Nets Hawk Katz. On the Erdős distinct distances problem in the plane. Ann. Math. (2), 181(1):155-190, 2015. URL: https://doi.org/10.4007/annals.2015.181.1.2.
18. Daniel M. Kane, Shachar Lovett, and Shay Moran. Near-optimal linear decision trees for k-SUM and related problems. J. ACM, 66(3):16:1-16:18, 2019. URL: https://doi.org/10.1145/3285953.
19. Jiří Matoušek. Range searching with efficient hierarchical cuttings. Discrete Comput. Geom., 10(2):157-182, 1993. URL: https://doi.org/10.1007/BF02573972.
20. Franco P. Preparata and Michael Ian Shamos. Computational Geometry - An Introduction. Texts and Monographs in Computer Science. Springer, 1985. URL: https://doi.org/10.1007/978-1-4612-1098-6.
21. Orit E. Raz, Micha Sharir, and Frank de Zeeuw. Polynomials vanishing on Cartesian products: the Elekes-Szabó theorem revisited. Duke Math. J., 165(18):3517-3566, 2016 . URL: https://doi.org/10.1215/00127094-3674103.
22. Orit E. Raz, Micha Sharir, and József Solymosi. Polynomials vanishing on grids: the Elekes-Rónyai problem revisited. Amer. J. Math., 138(4):1029-1065, 2016. URL: https://doi.org/10.1353/ajm.2016.0033.
23. József Solymosi and Frank de Zeeuw. Incidence bounds for complex algebraic curves on Cartesian products. In New Trends in Intuitive Geometry, volume 27 of Bolyai Soc. Math. Stud., pages 385-405. János Bolyai Math. Soc., Budapest, 2018.
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