Algebraic Integers Research Articles

An approach for computing generators of class fields of imaginary quadratic number fields using the Schwarzian derivative

Let N N be one of the 38 38 distinct square-free integers such that the arithmetic group Γ 0 ( N ) + \Gamma _0(N)^+ has genus one. We constructed canonical generators x N x_N and y N y_N for the associated function field (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 25 (2016), pp. 295–319]). In this article we study the Schwarzian derivative of x N x_N , which we express as a polynomial in y N y_N with coefficients that are rational functions in x N x_N . As a corollary, we prove that for any point e e in the upper half-plane which is fixed by an element of Γ 0 ( N ) + \Gamma _0(N)^+ , one can explicitly evaluate x N ( e ) x_N(e) and y N ( e ) y_N(e) . As it turns out, each value x N ( e ) x_N(e) and y N ( e ) y_N(e) is an algebraic integer which we are able to understand in the context of explicit class field theory. When combined with our previous article (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 29 (2020), pp. 1–27]), we now have a complete investigation of x N ( τ ) x_N(\tau ) and y N ( τ ) y_N(\tau ) at any CM point τ \tau , including elliptic points, for any genus one group Γ 0 ( N ) + \Gamma _0(N)^+ . Furthermore, the present article when combined with the two aforementioned papers leads to a procedure which we expect to yield generators of class fields, and certain subfields, using the Schwarzian derivative and which does not use either modular polynomials or Shimura reciprocity.

Unit group structure of the quotient ring of a quadratic ring

Let [Formula: see text] be the rational filed. For a square-free integer [Formula: see text] with [Formula: see text], we denote by [Formula: see text] the quadratic field. Let [Formula: see text] be the ring of algebraic integers of [Formula: see text]. In this paper, we completely determine the unit group of the quotient ring [Formula: see text] of [Formula: see text] for an arbitrary prime [Formula: see text] in [Formula: see text], where [Formula: see text] has the unique factorization property, and [Formula: see text] is a rational integer.

The Reciprocal Algebraic Integers Having Small House

Let α be a reciprocal algebraic integer of degree d. The house of α is the largest modulus of its conjugates. We compute the minimum of the houses of all reciprocal algebraic integers of degree d having the minimal polynomial which is a factor of a Dth degree reciprocal or antireciprocal polynomial with at most eight monomials, say , for d at most 180, and . We show that it is not necessary to take into account imprimitive polynomials. The computations suggest several conjectures. We show that d-th power of the house of a sequence of reciprocal primitive polynomials has a limit. We present a property of antireciprocal hexanomials.

Integral geometric orbifolds

A group of matrices with entries in a number field [Formula: see text] is integral if it has a finite index subgroup of matrices whose entries are algebraic integers. In this paper, we show that the fundamental groups, [Formula: see text], of the orbifolds [Formula: see text] are integral as subgroups of both [Formula: see text] and of [Formula: see text], for all the rational knots and links [Formula: see text] and all the isotropies with [Formula: see text] [Formula: see text]. We obtain the same result for the fundamental group [Formula: see text] of the orbifold [Formula: see text], [Formula: see text], where [Formula: see text] are the Borromean rings. The only groups [Formula: see text] with [Formula: see text], which are integral subgroups of both [Formula: see text] and of [Formula: see text] are for the following [Formula: see text]: [Formula: see text]

Alphabets, rewriting trails and periodic representations in algebraic bases

For \(\beta > 1\) a real algebraic integer (the base), the finite alphabets \({\mathcal {A}} \subset {\mathbb {Z}}\) which realize the identity \({\mathbb {Q}}(\beta ) = \mathrm{Per}_{{\mathcal {A}}}(\beta )\), where \(\mathrm{Per}_{{\mathcal {A}}}(\beta )\) is the set of complex numbers which are \((\beta , {\mathcal {A}})\)-eventually periodic representations, are investigated. Comparing with the greedy algorithm, minimal and natural alphabets are defined. The natural alphabets are shown to be correlated to the asymptotics of the Pierce numbers of the base \(\beta \) and Lehmer’s problem. The notion of rewriting trail is introduced to construct intermediate alphabets associated with small polynomial values of the base. Consequences on the representations of neighbourhoods of the origin in \({\mathbb {Q}}(\beta )\), generalizing Schmidt’s theorem related to Pisot numbers, are investigated. Applications to Galois conjugation are given for convergent sequences of bases \(\gamma _s := \gamma _{n, m_1 , \ldots , m_s}\) such that \(\gamma _{s}^{-1}\) is the unique root in (0, 1) of an almost Newman polynomial of the type \(-1+x+x^n +x^{m_1}+\cdots + x^{m_s}\), \(n \ge 3\), \(s \ge 1\), \(m_1 - n \ge n-1\), \(m_{q+1}-m_q \ge n-1\) for all \(q \ge 1\). For \(\beta > 1\) a reciprocal algebraic integer close to one, the poles of modulus \(< 1\) of the dynamical zeta function of the \(\beta \)-shift \(\zeta _{\beta }(z)\) are shown, under some assumptions, to be zeroes of the minimal polynomial of \(\beta \).

Non-negative integral matrices with given spectral radius and controlled dimension

Abstract A celebrated theorem of Douglas Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number p, we prove that there is an integral irreducible matrix with spectral radius p, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number p, there is an irreducible shift of finite type with entropy $\log (p)$ defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data.

Value-distribution of quartic Hecke L-functions

Set $K=\mathbb{Q}(i)$ and suppose that $c\in \mathbb{Z}[i]$ is a square-free algebraic integer with $c\equiv 1 \imod{\langle16\rangle}$. Let $L(s,\chi_{c})$ denote the Hecke $L$-function associated with the quartic residue character modulo $c$. For $\sigma>1/2$, we prove an asymptotic distribution function $F_{\sigma}$ for the values of the logarithm of \begin{equation*} L_c(s)= L(s,\chi_c)L(s,\overline{\chi}_{c}), \end{equation*} as $c$ varies. Moreover, the characteristic function of $F_{\sigma}$ is expressed explicitly as a product over the prime ideals of $\mathbb{Z}[i]$.

Dedekind's criterion for the monogenicity of a number field versus Uchida's and Lüneburg's

We compare three different characterizations, due respectively to R. Dedekind, K. Uchida, and H. Lüneburg, of when, given an algebraic integer, the ring generated by it coincides with the ring of integers of its quotient field, and apply our results to some concrete 2-towers of number fields.

House of algebraic integers symmetric about the unit circle

We give a Schinzel-Zassenhaus-type lower bound for the maximum modulus of roots of a monic integer polynomial with all roots symmetric with respect to the unit circle. Our results extend a recent work of Dimitrov, who proved the general Schinzel-Zassenhaus conjecture by using the Pólya rationality theorem for a power series with integer coefficients, and some estimates for logarithmic capacity (transfinite diameter) of sets. We use an enhancement of Pólya's result obtained by Robinson, which involves Laurent-type rational functions with small supremum norms, thereby replacing the logarithmic capacity with a smaller quantity. This smaller quantity is expressed via a weighted Chebyshev constant for the set associated with Dimitrov's function used in Robinson's rationality theorem. Our lower bound for the house confirms a conjecture of Boyd.

Sums of binomial coefficients evaluated at [formula omitted], and applications

The additive monoid R+(x) is defined as the set of all nonnegative integer linear combinations of binomial polynomials (xn) for n∈Z+. This paper is concerned with the inquiry into the structure of R+(α) for complex numbers α. Particularly interesting is the case of algebraic α which are not non-negative integers. This question is motivated by the study of functors between Deligne categories Rep(St) (and also Rep(GLt)) for t∈C﹨Z+. We prove that this object is a ring if and only if α is an algebraic number that is not a nonnegative integer. Furthermore, we show that all algebraic integers generated by α, i.e. all elements of OQ(α), are also contained in this ring. We also give two explicit representations of R+(α) for both algebraic integers and general algebraic numbers α. One is in terms of inequalities for the valuations with respect to certain prime ideals and the other is in terms of explicitly constructed generators. We show how these results work in the context of the study of symmetric monoidal functors between Deligne categories in positive characteristic. Moreover, this leads to a particularly simple description of R+(α) for both quadratic algebraic numbers and roots of unity.

A variant S2 measure for algebraic integers all of whose conjugates lie in a sector

Let α be a nonzero algebraic integer of degree d with conjugates α1=α,…,αd. It is well known that Sk(α)= ∑i=1dαik. Here, we define Sk′(α)= ∑i=1d|αi|k and sk′(α)=Sk′(α)∕d. Then, we focus our attention on the case k=2 when α is an algebraic integer all of whose conjugates lie in a sector |argz|≤𝜃, 0<𝜃<π2. We compute the greatest lower bound c(𝜃) of s2′(α) for 𝜃 belonging to eleven subintervals of [0,π2). Moreover, among these subintervals, there are twice two consecutive and complete subintervals. We use the method of explicit auxiliary functions for which the involved polynomials are found by our recursive algorithm.

Mahler measures of Pisot and Salem type numbers

In this paper we consider the set of Mahler measures of generalized Pisot and Salem numbers and give a characterization of such numbers in terms of their minimal polynomials. We show that every real quadratic algebraic integer β can be expressed by the difference of the Mahler measure of a generalized Pisot number and a Pisot number in ℚ(β).

Computing Garsia entropy for Bernoulli convolutions with algebraic parameters * *Research of K G Hare was supported by NSERC Grant 2019-03930.

We introduce a parameter space containing all algebraic integers β ∈ (1, 2] that are not Pisot or Salem numbers, and a sequence of increasing piecewise continuous function on this parameter space which gives a lower bound for the Garsia entropy of the Bernoulli convolution ν β . This allows us to show that dimH(ν β ) = 1 for all β with representations in certain open regions of the parameter space.

Totally positive algebraic integers with small trace

The “Schur-Siegel-Smyth trace problem” is a famous open problem that has existed for nearly 100 years. To study this problem with the known methods, we need to find all totally positive algebraic integers with small trace. In this work, on the basis of the classical algorithm, we construct a new type of explicit auxiliary functions related to Chebyshev polynomials to give better bounds for S k S_k , and reduce sharply the computing time. We are then able to push the computation to degree 15 15 and prove that there is no such totally positive algebraic integer with absolute trace 1.8 1.8 . As an application, we improve the lower bound for the absolute trace of totally positive algebraic integers to 1.793145 ⋯ 1.793145\cdots .

Complete first-order theories of some classical matrix groups over algebraic integers

Let O be the ring of integers of a number field, and let n≥3. This paper studies bi-interpretability of the ring of integers Z with the special linear group SLn(O), the general linear group GLn(O) and the subgroup Tn(O) of GLn(O) consisting of all the uppertriangular matrices. For each of these groups we provide a complete characterization of arbitrary models of their complete first-order theories.

Relative dynamical degree of monomial maps

We compute all relative dynamical degrees of equivariant dominant rational maps on toric varieties. We use the intersection theory on toric varieties using Minkowski weights. As a result, we see that the relative dynamical degrees can be reduced to the dynamical degrees on toric varieties. Hence these are all algebraic integers.

A sequence of algebraic integer relation numbers which converges to 4

Let α∈R and letA=[1101]andBα=[10α1]. The subgroup Gα of SL2(R) is a group generated by the matrices A and Bα. In this paper, we investigate the property of the group Gα. We construct a generalization of the Farey graph for the subgroup Gα. This graph determines whether the group Gα is a free group of rank 2. More precisely, the group Gα is a free group of rank 2 if and only if the graph is tree. In particular, we show that if 1/2 is a vertex of the graph, then Gα is not a free group of rank 2. Using this, we construct a sequence of real numbers so that the sequence converges to 4 and each number has the corresponding group that is not a free group of rank 2. It turns out that the real numbers are algebraic integers.

Chains of prime ideals in power series rings

An ideal I of a commutative ring D with identity is called an SFT ideal if there exist a finitely generated ideal J with J ⊆ I and a positive integer k such that a k ∈ J for each a ∈ I . We prove that for a non-SFT maximal ideal M of an integral domain D , ht ( M 〚 X 〛 / M D 〚 X 〛 ) ≥ 2 ℵ 1 if either (1) D is a 1-dimensional quasi-local domain (in particular D is a 1-dimensional nondiscrete valuation domain) or (2) M is the radical of a countably generated ideal. In other words, if one of the conditions (1) and (2) is satisfied, then there is a chain of prime ideals in D 〚 X 〛 with length at least 2 ℵ 1 such that each prime ideal in the chain lies between M D 〚 X 〛 and M 〚 X 〛 . As an application, assuming the continuum hypothesis we show that if D is either the ring of algebraic integers or the ring of integer-valued polynomials on Z , then dim ⁡ D 〚 X 〛 = ht M 〚 X 〛 = ht ( M 〚 X 〛 / M D 〚 X 〛 ) = 2 ℵ 1 for every maximal ideal M of D .

Finiteness of lattice points on varieties F(y) = F(g(𝕏)) + r(𝕏) over imaginary quadratic fields

We construct affine varieties over \mathbb{Q} and imaginary quadratic number fields \mathbb{K} with a finite number of \alpha-lattice points for a fixed \alpha\in \mathcal{O}_\mathbb{K}, where \mathcal{O}_\mathbb{K} denotes the ring of algebraic integers of \mathbb{K}. These varieties arise from equations of the form F(y) = F(g(x_1,x_2,\ldots, x_k))+r(x_1,x_2\ldots, x_k), where F is a rational function, g and r are polynomials over \mathbb{K}, and the degree of r is relatively small. We also give an example of an affine variety of dimension two, with a finite number of algebraic integral points. This variety is defined over the cyclotomic field \mathbb{Q}(\xi_3)=\mathbb{Q}(\sqrt{-3}).

Integral trace form of extensions of degree pq

In this work, we present the integral trace form [Formula: see text] of a cyclic extension [Formula: see text] with degree [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are distinct odd primes, the conductor of [Formula: see text] is a square free integer, and [Formula: see text] belongs to the ring of algebraic integers [Formula: see text] of [Formula: see text]. The integral trace form of [Formula: see text] allows one to calculate the packing radius of lattices constructed via the canonical (or twisted) homomorphism of submodules of [Formula: see text].