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Asymptotic estimates for rational spaces on hypersurfaces in function fields
Author(s) -
Zhao Xiaomei
Publication year - 2012
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdr031
Subject(s) - mathematics , hypersurface , rational function , polynomial , finite field , field (mathematics) , degree (music) , combinatorics , ring (chemistry) , function (biology) , polynomial ring , asymptotic formula , function field , discrete mathematics , pure mathematics , mathematical analysis , evolutionary biology , biology , chemistry , physics , organic chemistry , acoustics
Let q [ t ] denote the polynomial ring over the finite field q . For c 1 , …, c s ∈ q [ t ]\{0}, we consider the hypersurface H : c 1 z 1 k + … + c s z s k = 0. Let I P denote the subset of q [ t ] containing all polynomials of degree strictly less than P . Let N s , k , d , c ( P ) denote the number of d ‐tuples (x 1 ,… ,x d ) ∈ ( I P s ) d such that z = 1 x 1 +…+ d x d lies on H for all 1 , …, d ∈ q ( t ). In this paper, we apply a variant of the Hardy–Littlewood circle method to establish an asymptotic formula for N s , k , d , c ( P ). As a consequence, we see that under certain conditions there are infinitely many d ‐dimensional rational spaces on H .