Computational SectionIndecomposable integers in real quadratic fields
Introduction
The ring of algebraic integers of number field K is one of the key objects studied in algebraic number theory, and its additive structure sometimes plays a surprisingly important role. For example, in totally real number fields, we can focus on the semiring of totally positive elements of , denoted by , and define the subset of so-called indecomposable integers, i.e., elements of which cannot be expressed as a sum of two elements of . Several interesting applications of indecomposable integers to universal quadratic forms (i.e., positive quadratic forms over that represent all elements of ) have been recently developed by Blomer, Kala or Kim [1], [7], [2], [8], although they have been also used by Siegel [11] already in 1945 in a similar context.
All indecomposable integers in real quadratic fields , where is a squarefree integer, can be nicely described using the continued fraction expansion of or in the cases or , resp. – see Lemma 5 below.
Currently, we do not have such a characterization for the fields of higher degrees. Nevertheless, we can mention a partial result given by Čech, Lachman, Svoboda, Zemková and the first author [3] considering biquadratic fields, which have degree 4.
Using the above description for real quadratic fields, Dress and Scharlau [4, Theorem 3] deduced an upper bound on the norm of quadratic indecomposable integers. This result was later refined by Jang and Kim by proved the following result.
Theorem 1 [5] Theorem 5 Let be a squarefree integer and let −N be the largest negative norm of the algebraic integers in . Then for all indecomposable .
It is important to mention that equality always holds in this result if the norm of the fundamental unit in K is equal to −1. Moreover, except for the mentioned cases, this bound is lower than the bound given in [4].
In the same paper, Jang and Kim also stated a conjecture improving their upper bound.
Conjecture 2 [5] Let D and N be as in Theorem 1 and a be the smallest nonnegative rational integer such that N divides . Then for all indecomposable .
This conjecture was disproved by Kala [6, Theorem 4] in the case by giving a counterexample.
In this paper we study the norms of indecomposable integers in greater detail. In particular, we find the minimal D in each congruence class that provides a counterexample to the Jang-Kim Conjecture; provide infinite families of such counterexamples; and state a refined version of the Jang-Kim Conjecture (see Conjecture 3 below). If true, our families of examples show that our refined conjecture is best-possible.
Conjecture 3 Let D and N be as in Theorem 1. Let a be the smallest nonnegative rational integer such that if and such that if . Then for all indecomposable .
Notice that in addition to the extra term we have added, we have also corrected the main term for , as the definition of a in Jang-Kim's formulation of their conjecture in that case does not seem to fit with their expectation about where indecomposable integers of maximum norm occur (see their Proposition 1 as well as the paragraph before Conjecture 3 of [6]).
We have also proved the following result in the direction of our refined conjecture; namely that the Jang-Kim Conjecture is only wrong by at most .
Theorem 4 Let a, D and N be as in Conjecture 3. Then for all indecomposable .
Section 2 is devoted to study of indecomposable integers in where is a squarefree integer. We will introduce some notation as well as basic facts about continued fraction expansions and about algebraic integers in the real quadratic fields. In Sections 3 and 4, we provide the estimates required to prove Theorem 4. Section 5 contains the calculations which resulted in finding counterexamples for . Moreover, our search is exhaustive so the counterexamples provided here have minimal discriminants. Sections 6 and 7 contain details about the minimal counterexamples and the families of counterexamples found, while in Section 8 we present evidence suggesting that Conjecture 3 is best-possible as well as the proof of Theorem 4.
Section snippets
Preliminaries
Let K be a real quadratic number field and let denote the conjugate of . Let R be an order of discriminant in K. There exists a non-square positive integer D such that and either or . For the most part we will work with , but since the results of [4] are stated for any order, R, we do so initially here too.
Any is said to be totally positive, denoted by , if both α and are positive. We write for the set of such elements. We call α
Approximations of the 's
In this section, we will be concerned with determining the values of . We start with a recurrence relation for the norms . This result is a generalization of Proposition 5c) in [6].
Lemma 7 For any squarefree integer and for each , we have
Proof The main idea of the proof is based on the definitions of and and two relations given by Lemma 6. Using its definition, we can express as We know that and
Approximations of the norms of indecomposable integers
In the following lemma, we will determine when takes its largest value for a fixed index i.
Lemma 9 Let i be odd and be such that has the maximal value among where . If is even, then . If is odd, then is one of .
Proof We first express in terms of and . From the definitions and (2.3), we have
Computational work
Initially, the techniques developed by Kala [6] for were adapted for use with too. In this way, we found counterexamples to the conjecture of Jang and Kim for , as well as a smaller counterexample for than provided in [6]. For example, in this way, it was shown that yields a counterexample to the conjecture of Jang and Kim.
During this stage of the work, we directly examined small values of D. In this way, we found much smaller
The smallest for which we found a counterexample was . Here the continued fraction expansion of ξ is which has minimal period length 16. We obtain N from , for which , . , . For such D and N, we have , so the conjectured Jang-Kim upper bound is 487.
However, for , we have , , , which exceeds the Jang-Kim upper
Infinite families
As well as finding the smallest D for which the conjecture of Jang and Kim fails, we also found infinitely many examples for which this conjecture fails. Moreover, we also found families where the difference between the maximum norm of an indecomposable integer and the conjectured upper bound grows arbitrarily large.
Our interest in the particular families of counterexamples in Section 7 for is that if we let , then we find that from below. We know of no counterexamples where exceeds 1/8. None were found from our calculations described in Section 5 (as noted above, the largest value of that we found for was for ). Furthermore, all the examples found with large came from relatively short periods (at most of length 34), so we believe it
Acknowledgments
The authors gratefully acknowledge the many helpful suggestions of Vítězslav Kala during the preparation of this paper.
The first author was supported by Czech Science Foundation (GAČR), grant 17-04703Y, by the Charles University, project GA UK No. 1298218, by Charles University Research Centre program UNCE/SCI/022, and by the project SVV-2017-260456.
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