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Control and Cybernetics

Tytuł artykułu

Algorithms for integral solutions of a class of diophantine equations

Autorzy Polak, A. 
Treść / Zawartość
Warianty tytułu
Języki publikacji EN
EN In 1970 a negative solution to the tenth Hilbert problem, concerning the determination of integral solutions of diophantine equations, has been published by Y. W. Matiyasevich (see Matiyasevich, 1970). Despite this result, we can present algorithms to compute integral solutions (roots) for a wide class of quadratic diophantine equations of the form q(x) = d, where q : Zn→ Z is a homogeneous quadratic form. We will focus on the roots of one (i.e., d = 1) of quadratic Euler forms of selected posets from Loupias list (see Loupias, 1975). In particular, we will describe the roots of positive definite quadratic forms and the roots of quadratic forms that are principal (see Simson, 2010a). The algorithms and results we present here are successfully used in the representation theory of finite groups and algebras.
Słowa kluczowe
EN integral quadratic form   unit form   diophantine equations   roots   Euler bilinear form   Euclidean diagrams   mesh quiver   algorithm   Maple  
Wydawca Systems Research Institute, Polish Academy of Sciences
Czasopismo Control and Cybernetics
Rocznik 2011
Tom Vol. 40, no 2
Strony 491--514
Opis fizyczny Bibliogr. 16 poz.
autor Polak, A.
  • Faculty of Mathematics and Computer Science, Nicolaus Copernicus University ul. Chopina 12/18, 87-100 Toruń, Poland,
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Loupias, M. (1975) Indecomposable representations of finite partially ordered sets. Lecture Notes in Math., 488, Springer-Verlag, Berlin-Heidelberg-New York, 201-209.
Marczak,M., Polak,A. and Simson,D. (2010) P-critical integral quadratic forms and positive unit forms: An algorithmic approach. Linear Algebra and its Applications, 433, 1873-1888.
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