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On arithmetic graphs associated with integral domains. (English) Zbl 0727.11039
A tribute to Paul Erdős, 207-222 (1990).
[For the entire collection see Zbl 0706.00007.]
Let K be a finitely generated extension field of \({\mathbb{Q}}\), R a subring of K containing 1, S a finitely generated subgroup of the unit group of R with -1\(\in S\) and N a finite nonempty subset of \(R\setminus \{0\}\). For \(A=\{\alpha_ 1,...,\alpha_ m\}\subset R\), G(A) denotes the graph with vertex set A whose edges are the pairs \([\alpha_ i,\alpha_ j]\) with \(\alpha_ i-\alpha_ j\not\in \vartheta_{ij}S\), where \(\vartheta_{ij}\in N\). The subsets A and \(A'\) of R are called S- equivalent if \(A'=\epsilon A+\beta\) for some \(\epsilon\in S\) and \(\beta\in R\). Let \(\overline{G(A)}\) and \(\overline{G(A)}^ 0\) denote the complement of G(A) and the polygon hypergraph of \(\overline{G(A)}\), respectively.
The main result of this paper states: Let \(m\geq 3\). Then there exists a constant C(m,S) such that for all but at most C(m,S) S-equivalence classes of ordered subsets A of R, one of the following cases holds:
a) G(A) is connected and at least one of \(\overline{G(A)}\) and \(\overline{G(A)}^ 0\) is not connected,
b) G(A) has two connected components, \(G_ 1\) and \(G_ 2\), say, such that \(\overline{G_ 1}\) is not connected and \(| G_ 2| =1.\)
Furthermore, if \(m=4,\)
c) G(A) has two connected components of order 2 and \(\overline{G(A)}^ 0\) is not connected.
This result is applied to prove a bound for the numbers of solutions of systems of unit equations, and the finiteness of the number of solutions of resultant form equations. Further applications, e.q. to irreducible polynomials and decomposable form equations are announced.

11R04 Algebraic numbers; rings of algebraic integers
05C40 Connectivity
11D57 Multiplicative and norm form equations