DIVISORS IN A DEDEKIND DOMAIN

Javier Cilleruelo and Jorge Jiménez-Urroz

1 Introduction

Let A be a Dedekind domain in which we can define a notion of distance. We are interested in the number of divisors that an element α ∈ A can have in a small interval.

Of course, we can not talk about divisors if our domain is not a UFD. Hence, the convinient object to deal with will be the ideals of A.

We are able to prove a general theorem about the minimal length that an interval can have containing k divisors of a fixed M in terms of the size of these divisors.

Let φ be a real-valued function over the ideals in A such that

φ(α) ≥ 0 (i) φ(αβ) = φ(α) + φ(β), (ii)

and let us write (α, β) for the greatest common divisor of α and β. We prove

Theorem 1.1. Let α_{1}, ⋯ α_{k} be ideals in A with least common multiple M = [α_{1}, ⋯, α_{k}] and such that φ (α_{i}) ≥ γφ(N) for some multiple N of M. If L ∈ ℝ is such that φ((α_{j}, α_{i})) ≤ L for all 1 ≤ j, i ≤ k, then

L ≥ E_{k}(γ)φ(N)

where

E_{k}(γ) = \frac{[kγ](2kγ − [kγ] − 1)}{k(k − 1)}.

Notice that E_{k}(γ) > γ^{2} − γ (\frac{1−γ}{k−1} ) and that it is increasing as a function of either γ or k.

Typeset by AMS-TEX

Mathematics Subject Classification, 11R04, 11A05

This result came out when studying lattice points on conics, and this is in fact our principal application of the theorem.

By means of the identity N = (x+y\sqrt{d})(x−y\sqrt{d}), valid for any lattice point, (x, y), on the conic, and introducing the quadratic field ℚ (\sqrt{d}), all the problems about lattice points can be translated in terms of divisors (x + y\sqrt{d}) = α ∈ A of N, where A is the ring of integers of ℚ (\sqrt{d}).

So, one first problem that appears is trying to deal with ”divisors” when we are not in a unique factorization domain. This can be avoided by introducing the ideals in A, which guarantee unique factorization. The difficulty will now be how to translate the information from ideals to the elements.

In [2] and [3], this is only possible for principal ideals. We will get information from all the ideals, by means of Theorem 1.1 and noting that in fact one ideal is, in some sense, a divisor of its elements.

In this way, we give a new proof of Theorem 1 of [1], and give improvements on the principal results of [2] and [3].

Theorem 1.2. Let d ≠ 0, 1 be a fixed squarefree integer. On the conic, x^{2} − dy^{2} = N, an arc of length N^{α} with α ≤ 1/4 − 1/ (8 [k/2] + 4) contains at most k lattice points.

In this theorem we have avoided the case d = 1, considered in [4]. However, in this case we are able to prove the analogous result, but this time we will cover all the ranges of the hyperbola. Meanwhile as we have seen, Theorem 1.2 only includes γ = 1/2 of Theorem 1.1.

The key point for the improvement in this particular case is that any lattice point on x^{2} − y^{2} = N, gives us another one on the hyperbola XY = N, with coordinates X = x − y, Y = x + y. So, looking at the latter curve, we see that each lattice point corresponds to an integral divisor X ∈ ℤ of N. We can prove

Theorem 1.3. On the hyperbola xy = N there are at most k lattice points (x_{1}, y_{1}), ⋯, (x_{k}, y_{k}) such that N^{γ} ≤ x_{1} < ⋯ < x_{k} and x_{k} − x_{1} ≤ N^{E_{k}(γ)}.

Finally, in order to show the more general character of Theorem 1.1, we will include an application concerning polynomials.

Theorem 1.4. Let F_{1}(x), ⋯, F_{k}(x) be polynomials in ℤ[x] with least common multiple M(x) and such that deg(F_{i}(x)) ≥ γ deg(M(x)). Then, there exist i < j such that

deg(F_{j}(x) − F_{i}(x)) ≥ deg(M(x))E_{k}(γ).

Acknowledgment. We wish to thank A. Granville for his suggestion to state Theorem 1.1 in terms of ideals directly.

2 Proofs of theorems

Proof of theorem 1.1.

For any ideal β ∈ A and some prime ideal π, we define v_{π} (β) = t to be the greatest power of π dividing β.

v_{π} is well defined since, in a Dedekind domain, we have unique factorization of ideals. Further, we know every ideal α has an inverse α^{−1} which is a fractional ideal. Hence, we can extend the definition of v_{π} and φ to the inverses of ideals in such a way that v_{π} (α^{−1}) = −v_{π} (α), and φ (α^{−1}) = −φ (α).

Now, let us order the ideals α_{1}, ⋯, α_{k} such that v_{π} (α_{i}) = t_{i} increases with i. Then

v_{π} (∏(α_{j}, α_{i})) = ∑_{1≤i<j≤k} t_{i} = ∑_{i=1}^{k−1} t_{i} ∑_{j=i+1}^{k} 1 = ∑_{i=1}^{k} t_{i}(k − i),

and on the other hand

v_{π} (∏ α_{i}) = ∑_{1≤i≤k} t_{i}.

Hence, grouping all the local information of each prime, we can write

∏(α_{j}, α_{i}) = ∏_{π|M} π^{∑_{1≤i≤k} t_{i}(k−i)} ∏ α_{i} = ∏_{π|M} π^{∑_{1≤i≤k} t_{i}},

and so, for any integer m we have

(2.1)  ∏(α_{j}, α_{i}) = (∏ α_{i})^{m} ∏_{π|M} π^{∑_{1≤i≤k} t_{i}(k−i−m)}.

Now, since k − i − m ≥ 0 when i ≤ k − m, we have

∑_{1≤i≤k} t_{i}(k − i − m) ≥ − ∑_{k−m≤i≤k} t_{i}(i − (k − m)) ≥ −t_{k} ∑_{k−m≤i≤k}(i − (k − m))
= − t_{k}\binom{m + 1}{2},

where we have used t_{i} ≤ t_{k}. Hence, by properties (i) and (ii) of φ and looking at the identity M = ∏_{π|M} π^{t_{k}}, we deduce by substitution in (2.1) that

\binom{k}{2}L ≥ ∑ φ ((α_{j}, α_{i})) ≥ m ∑ φ (α_{i}) − \binom{m + 1}{2}φ (M) ,

and so, from the hypothesis in the theorem φ (α_{i}) ≥ γφ (N), and φ (M) ≤ φ (N), (since M|N), we get

\binom{k}{2}L ≥ φ (N) (kγm − \binom{m + 1}{2}) .

The proof is now concluded by choosing m = [kγ], which maximizes the previous quantity.

Proof of Theorem 1.4.

Let us first prove Theorem 1.4, and see how Theorem 1.1 works in that context. So, consider A = ℤ[x]. This is a principal ideal domain and the function φ (F) = deg F (x), which has properties (i) and (ii), is well defined, where F (x) is the generator of the ideal F. The conclusion of the theorem is now clear since deg(F_{j}(x) − F_{i}(x)) ≥ deg(F_{j}(x), F_{i}(x)).

Proof of Theorem 1.2.

Suppose we now have k lattice points (a_{1}, b_{1}), ⋯, (a_{k}, b_{k}) on the conic x^{2} − dy^{2} = N, with d ≠ 0, 1. Let A be the ring of integers of the quadratic field ℚ(\sqrt{d}), α_{i} =< a_{i} + b_{i}\sqrt{d} > the ideal generated by a_{i} + b_{i}\sqrt{d} and consider, for any α ideal in A, the function φ (α) = log (N(α)), where N(α) is the norm of the ideal. This function again has the properties of Theorem 1.2.

Now, since a_{i}^{2} − db_{i}^{2} = N, we have N(α_{i}) = N, and we find < N > to be a multiple of the least common multiple of α_{i}. Now, φ (< N >) = 2 log N, and hence φ (α_{i}) = 1/2φ (< N >), so by Theorem 1.1

(2.2)  φ ((α_{j}, α_{i})) ≥ 2E_{k} (1/2) log N.

On the other hand, for any 1 ≤ i < j ≤ k we have clearly |ξ_{j}−ξ_{i}| ≥ \sqrt{N(ξ_{j} − ξ_{i})}, where ξ_{i} = a_{i} + b_{i}\sqrt{d}, and |ξ| is the euclidean distance from ξ to the origin O = 0 + 0\sqrt{d}, and so

log |ξ_{j} − ξ_{i}| ≥ \frac{1}{2} log (|N(ξ_{j} − ξ_{i})|) = \frac{1}{2}φ (< ξ_{j} − ξ_{i} >) .

Divisors in a Dedekind domain 5

Finally, we have < ξ_{j} − ξ_{i} >⊂< α_{j} − α_{i} >, and we know, [5], that < α_{j} − α_{i} >= (α_{j}, α_{i}), so (α_{j}, α_{i})| < ξ_{j} − ξ_{i} >, and by the properties of φ and (2.2),

2 log |ξ_{j} − ξ_{i}| ≥ φ (< ξ_{j} − ξ_{i} >) ≥ φ ((α_{j}, α_{i})) ≥ 2E_{k} (1/2) log N,

which ends the proof.

proof of Theorem 1.3.

To prove the case d = 1, or more concretely Theorem 1.3, we will apply Theorem 1.1 to A = ℤ, and φ(x) = log |x|, where x is an ideal or the element generating the ideal. So, Theorem 1.1 together with

φ (x_{j} − x_{i}) ≥ φ ((x_{j}, x_{i})) ,

gives the result.

References

[1] J. Cilleruelo and A. Córdoba, Trigonometric Polynomials and Lattice Points, Proc. of Amer. Math. Soc. 115.4 (1992), 899–905.

[2] J. Cilleruelo and A. Córdoba, Lattice Points on Ellipses, Duke Math. Jour. 76.3 (1994), 741–750.

[3] J. Cilleruelo and J. Jiménez-Urroz, Lattice points on Hyperbolas, J. Number Theory 63,2 (1997), 267–274.

[4] A. Granville and J. Jiménez-Urroz, The least common multiple and lattice points on hyperbolas, Preprint (1995).

[5] I.N. Stewart and D.O. Tall, Algebraic Number Theory, Chapman and Hall, London, 1987.

Departamento de Matemáticas, Facultad de Ciencias Universidad Autónoma de Madrid, 28049 Madrid, ESPAÑA.

e-mail: franciscojavier.cilleruelo@uam.es, jorge.jimenez@uam.es