Abstract
In Part I of the present paper the following problem was investigated. Let G be a finite simple graph, and S be a finite set of primes. We say that G is representable with S if it is possible to attach rational numbers to the vertices of G such that the vertices v 1, v 2 are connected by an edge if and only if the difference of the attached values is an S-unit. In Part I we gave several results concerning the representability of graphs in the above sense.
In the present paper we extend the results from Part I to the algebraic number field case and make some of them effective. Besides we prove some new theorems: we prove that G is infinitely representable with S if and only if it has a degenerate representation with S, and we also deal with the representability with S of the union of two graphs of which at least one is finitely representable with S.
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References
Amoroso F., Viada E.: Small points on subvariaties of a torus. Duke Math. J. 150, 407–442 (2009)
H. Cohen, Advanced Topics in Computational Number Theory Springer (2000).
A. Ćustić, L. Hajdu, D. Kreso and R. Tijdeman, On conjectures and problems of Ruzsa concerning difference graphs of S-units, Acta Math. Hungar. 146 (2015), 391–404.
Evertse J.-H.: On equations in S-units and the Thue-Mahler equation. Invent. Math. 78, 561–584 (1984)
Evertse J.-H.: The number of solutions of decomposable form equations. Invent. Math. 122, 559–601 (1995)
Evertse J.-H., Győry K.: On the number of solutions of weighted unit equations. Compositio Math. 66, 329–354 (1988)
J.-H. Evertse, K. Győry, C. L. Stewart and R. Tijdeman, On S-unit equations in two unknowns, Invent. Math. 92 (1988), 461–474.
J.-H. Evertse, H.-P. Schlickewei and W. M. Schmidt, Linear equations in variables which lie in a multiplicative group, Annals of Math. 155 (2002), 807–836.
K. Győry, On the number of solutions of linear equations in units of an algebraic number field, Comment. Math. Helv. 54 (1979), 583–600.
K. Győry, On certain graphs composed of algebraic integers of a number field and their applications, I, Publ. Math Debrecen 27 (1980), 229–242.
Győry K.: Some recent applications of S-unit equations. Astérisque 209, 17–38 (1992)
K. Győry, On certain arithmetic graphs and their applications to Diophantine problems, Funct. Approx. 39 (2008), 289–314.
K. Győry, L. Hajdu and R. Tijdeman, Representation of finite graphs as difference graphs of S-units, I, J. Combinatorial Theory, Ser A 127 (2014), 314–335.
K. Győry, and K. Yu, Bounds for the solutions of S-unit equations and decomposable form equations, Acta Arith. 123 (2006), 9–41.
Laurent M.: Équations diophantiennes exponentielles. Invent. Math. 78, 299–327 (1984)
Ruzsa I. Z.: The difference graph of S-units. Publ. Math. Debrecen 79, 675–685 (2011)
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Research was supported in part by the OTKA grants NK104208 (K.Gy.) and K100339, K115479 (K.Gy. and L.H.).
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Győry, K., Hajdu, L. & Tijdeman, R. Representation of finite graphs as difference graphs of S-units. II. Acta Math. Hungar. 149, 423–447 (2016). https://doi.org/10.1007/s10474-016-0633-y
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DOI: https://doi.org/10.1007/s10474-016-0633-y