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Abstract: A rational positive-definite quadratic form is perfect if it can bereconstructed from the knowledge of its minimal nonzero value m and the finiteset of integral vectors v such that fv = m. This concept was introduced byVoronoi and later generalized by Koecher to arbitrary number fields. One knowsthat up to a natural -change of variables- equivalence, there are onlyfinitely many perfect forms, and given an initial perfect form one knows how toexplicitly compute all perfect forms up to equivalence. In this paper weinvestigate perfect forms over totally real number fields. Our main resultexplains how to find an initial perfect form for any such field. We alsocompute the inequivalent binary perfect forms over real quadratic fieldsQ\sqrt{d} with d \leq 66.



Author: Paul E. Gunnells, Dan Yasaki

Source: https://arxiv.org/







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