Explicit estimates for the number of rational points of singular complete intersections over a finite field

JOURNAL OF NUMBER THEORY

卷:158

Article

Matera, Guillermo^1,2  Perez, Mariana¹  Privitelli, Melina³ 

[1] Univ Nacl Gen Sarmiento, Inst Desarrollo Humano, Buenos Aires, DF, Argentina

[2] Natl Council Sci & Technol CONICET, Buenos Aires, DF, Argentina

[3] Univ Nacl Gen Sarmiento, Inst Ciencias, Buenos Aires, DF, Argentina

关键词: Finite fields;    Singular complete intersections;    Rational points;    Bertini's smoothness theorem;    Hooley-Katz estimate;   

DOI  :  10.1016/j.jnt.2015.06.007

来源: Elsevier

【 摘 要 】

Let V subset of P-n((F) over bar (q)) be a complete intersection defined over a finite field F-q of dimension r and singular locus of dimension at most 0 <= s <= r - 2. We obtain an explicit version of the Hooley-Katz estimate vertical bar vertical bar V(F-q)vertical bar - p(r vertical bar) = O(q((r+s+1)/2)), where vertical bar V(F-q)vertical bar denotes the number of F-q-rational points of V and p(r) : = vertical bar p(r)(F-q)vertical bar. Our estimate improves all the previous estimates in several important cases. Our approach relies on tools of classical algebraic geometry. A crucial ingredient is a new effective version of the Bertini smoothness theorem, namely an explicit upper bound of the degree of a proper Zariski closed subset of (P-n)(s+1) ((F) over bar (q)) which contains all the singular linear sections of V of codimension s + 1. (C) 2015 Elsevier Inc. All rights reserved.

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