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Divisors in a Dedekind domain. (English) Zbl 0910.11043

Let \(R\) be a Dedekind domain, and let \(\varphi\) be an additive non-negative real function defined on the set of all ideals of \(R\). The authors show that if \(I_1,I_2,\dots,I_k\) are ideals of \(R\), \(\mathcal M\) is their least common multiple and for some multiple \(\mathcal N\) of \(\mathcal M\) one has \(\varphi(I_j)\geq\gamma\varphi({\mathcal N})\) (\(i=1,2,\dots,k)\) then \[ \max\{\varphi( \text{gcd}(I_j,I_k)): i\neq j\}\geq E_k(\gamma)\varphi({\mathcal N}), \] where \(E_k(\gamma)\) is explicitly given and increases as a function of any of its arguments. This result is then used to obtain information about lattice points on conics: it is shown (Theorem 1.2) that on an arc of length \(N^\alpha\) (with \(\alpha\leq 1/4-1/(8[k/2]+4)\)) on the conic \(X^2-dY^2=N\) (where \(d\neq 0,1\) is a fixed square-free integer) there can be at most \(k\) lattice points. This is an essential improvement of previous results in this direction [J. Cilleruelo and A. Cordoba, Duke Math. J. 76, 741-750 (1994; Zbl 0822.11069); J. Cilleruelo and J. Jiménez-Urroz, J. Number Theory 63, 267-274 (1997; Zbl 0877.11051)]. Moreover it is shown that on the hyperbola \(xy=N\) there are at most \(k\) lattice points lying above the interval \([N^\gamma,M]\) provided \(M-N^\gamma\leq N^{E_k(\gamma)}\). An application to polynomials with rational integral coefficients is also given.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11P21 Lattice points in specified regions
13A05 Divisibility and factorizations in commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
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