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On a Question of Mordell and a Spectrum of Linear Forms
Author(s) -
Burger Edward B.
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700001368
Subject(s) - unimodular matrix , mathematics , infimum and supremum , combinatorics , converse , real number , integer (computer science) , geometry of numbers , minkowski space , matrix (chemical analysis) , discrete mathematics , geometry , materials science , computer science , composite material , programming language
Let L 1 ( x ), L 2 ( x ), …, L N ( x ) be N real linear forms in N variables defined byL m ( x ) = ∑ n = 1 Nc m nx n ,m = 1 , 2 , … , N ,such that the associated N × N matrix of coefficients C = ( c mn ) is unimodular. In the classical theory of geometry of numbers, Minkowski's well‐known linear forms theorem asserts that for any positive numbers ɛ 1 , ɛ 2 , …, ɛ N satisfying ɛ 1 , ɛ 2 , …, ɛ N ⩾ 1, there exists a lattice point p ∈ Z N , p ≠ 0 , such that | L m ( p )|⩽ɛ m , for m = 1,2,…, N . (1.1) In 1937, Mordell [ 7 ] posed the following question which may be viewed, in some sense, as a converse to Minkowski's result. Does there exist a constant Δ N such that for each N × N unimodular real matrix C , there exist positive real numbers ɛ 1 , ɛ 2 , …, ɛ N satisfying ɛ 1 ɛ 2 …ɛ N =Δ N , such that the only integer solution to the inequalities of (1.1) is p = 0 ? Moreover, if Δ N does exist, what is the best possible value for Δ N , that is, what is the supremum of all such admissible values of Δ N ? Clearly, by Minkowski's theorem, if Δ N exists, then Δ N < 1.