24,223 research outputs found

    Spectral Factorization of Trigonometric Polynomials and Lattice Geometry

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    We formulate a conjecture concerning spectral factorization of a class of trigonometric polynomials of two variables and prove it for special cases. Our method uses relations between the distribution of values of a polynomial of two variables and the distributions of values of an associated family of polynomials of one variable. We suggest an approach to prove the full conjecture using relations between the distribution of values and the distribution of roots of polynomials.Comment: This paper has been accepted for publication in Acta Arithmetica, it is a thoroughly revised and retitled version of the arXiv:1110.5277 paper submitted on 24 Oct 2011 that was titled "Spectral Factorization and Lattice Geometry

    Generators, Relations and Symmetries in Pairs of 3x3 Unimodular Matrices

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    Denote the free group on two letters by F2 and the SL(3,C)-representation variety of F2 by R = Hom(F2, SL(3,C)). There is a SL(3,C)-action on the coordinate ring of R, and the geometric points of the subring of invariants is an affine variety X. We determine explicit minimal generators and defining relations for the subring of invariants and show X is a degree 6 hyper-surface in C9 mapping onto C8. Our choice of generators exhibit Out(F2) symmetries which allow for a succinct expression of the defining relations.Comment: 21 pages. This fourth version is the author's final version which includes new comments and some correction
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