Abstract
We give an explicit construction of a large subset \({S \subset \mathbb{F}^n}\), where \({\mathbb{F}}\) is a finite field, that has small intersection with any affine variety of fixed dimension and bounded degree. Our construction generalizes a recent result of Dvir and Lovett (STOC 2012) who considered varieties of degree one (that is, affine subspaces).
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References
A Ben-Aroya & I Shinkar (2012). A Note on Subspace Evasive Sets. Electronic Colloquium on Computational Complexity (ECCC) 19, 95.
Bourgain J. (2007) On the construction of affine extractors. Geometric And Functional Analysis 17(1): 33–57
Zeev Dvir & Shachar Lovett (2012). Subspace evasive sets. STOC 2012 (to appear).
Jordan S. Ellenberg, Richard Oberlin & Terence Tao (2010). The Kakeya set and maximal conjectures for algebraic varieties over finite fields. Mathematika 56, 1–25.
A. Gabizon & R. Raz (2008). Deterministic extractors for affine sources over large fields. Combinatorica 28, 415–440. ISSN 0209-9683. URL http://portal.acm.org/citation.cfm?id=1459886.145988.
Phillip Griffiths & Joe Harris (1985). On the Noether-Lefschetz theorem and some remarks on codimension-two cycles. Math. Ann. 271(1), 31–51. ISSN 0025-5831.
V. Guruswami (2011). Linear-Algebraic List Decoding of Folded Reed-Solomon Codes. Annual IEEE Conference on Computational Complexity 77–85. ISSN 1093-0159.
Guruswami V., Rudra A. (2008) Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy. IEEE Transactions on Information Theory 54(1): 135–150
Joos Heintz (1983). Definability and Fast Quantifier Elimination in Algebraically Closed Fields. Theor. Comput. Sci. 24, 239–277.
János Kollár (1992). Trento examples. In Classification of irregular varieties (Trento, 1990), volume 1515 of Lecture Notes in Math., 136–139. Springer, Berlin.
János Kollár, Lajos Rónyai & Tibor Szabó (1996). Norm-Graphs and Bipartite Turán Numbers. Combinatorica 16(3), 399–406.
I. R. Shafarevich (1994). Basic algebraic geometry. Springer-Verlag New York, Inc., New York, NY, USA. ISBN 0-387-54812-2.
Claire Voisin (1989). Sur une conjecture de Griffiths et Harris. In Algebraic curves and projective geometry (Trento, 1988), volume 1389 of Lecture Notes in Math., 270–275. Springer, Berlin. URL http://dx.doi.org/10.1007/BFb008593.
Claire Voisin (2003). Hodge theory and complex algebraic geometry. II, volume 77 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge. ISBN 0-521-80283-0, x+351. URL http://dx.doi.org/10.1017/CBO978051161517. Translated from the French by Leila Schneps.
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Dvir, Z., Kollár, J. & Lovett, S. Variety Evasive Sets. comput. complex. 23, 509–529 (2014). https://doi.org/10.1007/s00037-013-0073-9
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DOI: https://doi.org/10.1007/s00037-013-0073-9