Mesh Algorithms for Solving Principal Diophantine Equations, Sand-glass Tubes and Tori of Roots

• Autorzy: Simson D.
• Języki publikacji: EN

Abstrakty

EN

We study integral solutions of diophantine equations q(x) = d, where x = (x1, . . . , xn), n ≥1, d .∈Z is an integer and q : Z^n →Z is a non-negative homogeneous quadratic form. Contrary to the negative solution of the Hilbert’s tenth problem, for any such a form q(x), we give efficient algorithms describing the set Rq(d) of all integral solutions of the equation q(x) = d in a Φ_A-mesh translation quiver form. We show in Section 5 that usually the set Rq(d) has a shape of a Φ_A-mesh sand-glass tube or of a A-mesh torus, see 5.8, 5.10, and 5.13. If, in addition, the subgroup Ker q = {v ∈Z^n; q(v) = 0} of Zn is infinite cyclic, we study the solutions of the equations q(x) = 1 by applying a defect δ_A : Z^n → Z and a reduced Coxeter number čA ∈ N defined by means of a morsification b_A : Zn × Zn → Z of q, see Section 4. On this way we get a simple graphical algorithm that constructs all integral solutions in the shape of a mesh translation oriented graph consisting of Coxeter A-orbits. It turns out that usually the graph has at most three infinite connected components and each of them has an infinite band shape, or an infinite horizontal tube shape, or has a sand-glass tube shape. The results have important applications in representation theory of groups, algebras, quivers and partially ordered sets, as well as in the study of derived categories (in the sense of Verdier) of module categories and categories of coherent sheaves over algebraic varieties.

Słowa kluczowe

EN

principal quadratic form; quiver; poset; tubular mesh algorithm; Coxeter matrix; morsification; defect; reduced Coxeter number; sand-glass tube; mesh geometry

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2011
• Tom: Vol. 109, nr 4
• Strony: 425--462
• Opis fizyczny: Bibliogr. 25 poz., tab., wykr.

Twórcy

autor

Simson D.

• Faculty of Mathematics and Computer Science Nicolaus Copernicus University ul. Chopina 12/18, 87-100 Toru´n, Poland,

Bibliografia

• [1] I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras, Volume 1. Techniques of Representation Theory, London Math. Soc. Student Texts 65, Cambridge Univ. Press, Cambridge-New York, 2006.
• [2] M. Auslander, I. Reiten and S. Smalø, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, 1995.
• [3] M. Barot and J.A. de la Pe˜na, The Dynkin type of a non-negative unit form, Expo. Math. 17(1999), 339-348
• [4] V. M. Bondarenko and A. M. Polishchuck, On finiteness of critical Tits forms of posets, Proc. Inst. Math. NAS Ukraine 50(2004), 1061-1063.
• [5] V.M. Bondarenko and M. V. Stepochkina, (Min, max)-equivalency of partially ordered sets and Tits quadratic forms, In: Analysis and Algebra Problems, Inst. Mat. NAS Ukraine 2(3), 2005, pp. 3-46.
• [6] Z. I. Borevich and I. R. Shafarevich, Number Theory, Izd. Nauka, Moscow, 1964 (in Russian).
• [7] P. Gabriel and A. V. Roiter, Representations of Finite Dimensional Algebras, Algebra VIII, Encyclopaedia of Math. Sc., Vol. 73, Springer-Verlag, 1992.
• [8] M. Ga¸siorek and D. Simson, One-peak posets with positive Tits quadratic form, their mesh translation quivers of roots, and programming in Maple and Python, Linear Algebra Appl. 2011, to appear.
• [9] I. M. Gelfand and V. A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in a finite-dimensional vector space. Coll. Math. Soc. Bolyai, Tihany (Hungary), 5 (1970), 163-237.
• [10] H. J. von Höhne, On weakly positive unit forms, Comment. Math. Helvetici, 63(1988), 312-336.
• [11] H. Lenzing, Coxeter transformations associated with finite dimensional algebras, Progress in Math. Birkh¨asuer Verlag Basel 173(1999), pp. 287-308.
• [12] H. Lenzing and J.A. de la Pe˜na, Spectral analysis of finite dimensional algebras and singularities, In: Trends in Representation Theory of Algebras and Related Topics, ICRA XII, (ed. A. Skowroński), Series of Congress Reports, EuropeanMath. Soc. Publishing House, Zürich, 2008, pp. 541-588.
• [13] G. Marczak, A. Polak and D. Simson, P-critical integral quadratic forms and positive forms. An algorithmic approach, Linear Algebra Appl. 433(2010), 1873-1888, doi: 10.1016//j.laa. 2010.06.052.
• [14] J. V. Matijasevich, Enumerable sets are Diophantine, Doklady Akad. Nauk SSSR. 191(1970), 279-282 (in Russian).
• [15] S. A. Ovsienko, Integral weakly positive forms, in ,,Schur Matrix Problems and Quadratic Forms", Inst. Mat. Akad. Nauk USSR, Preprint 78.25 (1978), pp. 3 - 17 (in Russian).
• [16] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math., Vol. 1099, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1984.
• [17] M. Sato, Periodic Coxeter matrices and their associated quadratic forms, Linear Algebra Appl. 406(2005), 99-108; doi: 10.1016//j.laa. 2005.03.036.
• [18] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra, Logic and Applications, Vol. 4, Gordon & Breach Science Publishers, 1992.
• [19] D. Simson, Incidence coalgebras of intervally finite posets, their integral quadratic forms and comodule categories, Colloq. Math. 115(2009), 259-295.
• [20] D. Simson, Mesh geometries of root orbits of integral quadratic forms, J. Pure Appl. Algebra, 215(2011), 13-34; doi: 10.1016/j.jpaa. 2010.02.029.
• [21] D. Simson, Integral bilinear forms, Coxeter transformations and Coxeter polynomials of finite posets, Linear Algebra Appl. 433(2010), 699-717; doi: 10.1016/j.laa. 2010.03.04.
• [22] D. Simson, Algorithms determining matrix morsifications, Weyl orbits, Coxeter polynomials and mesh geometries of roots for Dynkin diagrams, preprint 2011.
• [23] D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras, Volume 2. Tubes and Concealed Algebras of Euclidean Type, London Math. Soc. Student Texts 71, Cambridge Univ. Press, Cambridge-New York, 2007.
• [24] D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras, Volume 3. Representation-Infinite Tilted Algebras, London Math. Soc. Student Texts 72, Cambridge Univ. Press, Cambridge-New York, 2007.
• [25] K. Szymiczek, Bilinear Algebra: An Introduction to the Algebraic Theory of Quadratic Forms, Algebra, Logic and Applications, Vol. 7, Gordon & Breach Science Publishers, 1997.