کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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4593530 | 1630656 | 2016 | 19 صفحه PDF | دانلود رایگان |
Let V⊂Pn(F‾q) be a complete intersection defined over a finite field FqFq of dimension r and singular locus of dimension at most 0≤s≤r−20≤s≤r−2. We obtain an explicit version of the Hooley–Katz estimate ||V(Fq)|−pr|=O(q(r+s+1)/2)||V(Fq)|−pr|=O(q(r+s+1)/2), where |V(Fq)||V(Fq)| denotes the number of FqFq-rational points of V and pr:=|Pr(Fq)|pr:=|Pr(Fq)|. 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 (Pn)s+1(F‾q) which contains all the singular linear sections of V of codimension s+1s+1.
Journal: Journal of Number Theory - Volume 158, January 2016, Pages 54–72