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