An analogue of the Korkin-Zolotarev lattice reduction for vector spaces over number fields

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Title: An analogue of the Korkin-Zolotarev lattice reduction for vector spaces over number fields
Author: Rothlisberger, Mark Peter
Abstract: We show the existence of a basis for a vector space over a number field with two key properties . First , the n -th basis vector has a small twisted height which is bounded above by a quantity involving the n -th successive minima associated with the twisted height . Second , at each place v of the number field , the images of the basis vectors under the automorphism associated with the twisted height satisfy near -orthogonality conditions analagous to those introduced by Korkin and Zolotarev in the classical Geometry of Numbers . Using this basis , we bound the Mahler product associated with the twisted height . This is the product of a successive minimum of a twisted height with the corresponding successive minimum of its dual twisted height . Previous work by Roy and Thunder in [12] showed that the Mahler product was bounded above by a quantity which grows exponentially as the dimension of the vector space increases . In this work , we demonstrate an upper bound that exhibits polynomial growth as the dimension of the vector space increases .
URI: http : / /hdl .handle .net /2152 /ETD -UT -2010 -08 -1834
Date: 2010-12-14

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An analogue of the Korkin-Zolotarev lattice reduction for vector spaces over number fields. Doctoral dissertation, University of Texas at Austin. Available electronically from http : / /hdl .handle .net /2152 /ETD -UT -2010 -08 -1834 .

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