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The Distribution of Reciprocal Pairs Modulo Polynomials over a Finite Field
Author(s) -
Hensley Doug
Publication year - 1997
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19971840108
Subject(s) - mathematics , kloosterman sum , finite field , modulo , combinatorics , polynomial , order (exchange) , distribution (mathematics) , degree (music) , inverse , reciprocal , ring (chemistry) , field (mathematics) , term (time) , polynomial ring , discrete mathematics , pure mathematics , mathematical analysis , geometry , linguistics , philosophy , physics , chemistry , organic chemistry , finance , acoustics , economics , quantum mechanics
Given a finite field F q of order q , a fixed polynomial g in – F q [ X ] of positive degree, and two elements u and v in the ring of polynomials in R = F q [ X ]/ gF q [ X ], the question arises: How many pairs (a, 6) are there in R × R so that ab 1 mod g and so that a is close to u while b is close to v ? The answer is, about as many as one would expect. That is, there are no favored regions in R × R where inverse pairs cluster. The error term is quite sharp in most cases, being comparable to what would happen with random distribution of pairs. The proof uses Kloosterman sums and counting arguments. The exceptional cases involve fields of characteristic 2 and composite values of g . Even then the error term obtained is nontrivial. There is no computational evidence that inverses are in fact less evenly distributed in this case, however.