Algebraic Number Research Articles

Four Variants of Riemann Zeta Function

By means of the generating function method and Dougall’s formulae for bilateral hypergeometric series, we examine four classes of infinite series, which may be considered as variants of Riemann zeta function. Several summation formulae are established in closed form, which shows remarkably that the values of these series result in multiples of integer powers of π either by rational numbers or by algebraic numbers.

A uniform bound on the multiplicity of Dirichlet eigenvalues in planar annuli

We study the multiplicity of Dirichlet eigenvalues in the planar annulus Ωr={x∈R2:r<|x|<1} when r is an algebraic number. Using the algebraic independence results of the Bessel functions developed by Siegel and Shildlovskii, we prove that all Dirichlet eigenvalues here have multiplicity at most 6.

The topological slice genus of satellite knots

This paper presents evidence supporting the surprising conjecture that in the topological category the slice genus of a satellite knot $P(K)$ is bounded above by the sum of the slice genera of $K$ and $P(U)$. Our main result establishes this conjecture for a variant of the topological slice genus, the $\mathbb{Z}$-slice genus. As an application, we show that the $(n,1)$-cable of any 3-genus 1 knot (e.g. the figure 8 or trefoil knot) has topological slice genus at most 1. Further, we show that the lower bounds on the slice genus coming from the Tristram-Levine and Casson-Gordon signatures cannot be used to disprove the conjecture. Notably, the conjectured upper bound does not involve the algebraic winding number of the pattern $P$. This stands in stark contrast with the smooth category, where for example there are many genus 1 knots whose $(n,1)$-cables have arbitrarily large smooth 4-genera.

Stewart’s theorem revisited: suppressing the norm $$\pm 1$$ hypothesis

Let \(\gamma\) be an algebraic number of degree 2 and not a root of unity. In this note we show that there exists a prime ideal \({\mathfrak {p}}\) of \({\mathbb {Q}}(\gamma )\) satisfying \(\nu _{\mathfrak {p}}(\gamma ^n-1)\ge 1\), such that the rational prime p underlying \({\mathfrak {p}}\) grows quicker than n.

Algebraic Number Starscapes

We study the geometry of algebraic numbers in the complex plane, and their Diophantine approximation, aided by extensive computer visualization. Motivated by the resulting images, which we have called algebraic starscapes, we describe the geometry of the map from the coefficient space of polynomials to the root space, focusing on the quadratic and cubic cases. The geometry describes and explains the notable features of the illustrations, and motivates a geometric-minded recasting of fundamental results in the Diophantine approximation of the complex plane. Meanwhile, the images provide a case-study in the symbiosis of illustration and research, and an entry-point to geometry and number theory for a wider audience. In particular, the paper is written to provide an accessible introduction to the study of homogeneous geometry and Diophantine approximation. We investigate the homogeneous geometry of root and coefficient spaces under the natural action. Hyperbolic geometry and the discriminant play an important role in low degree. In particular, we rediscover the quadratic and cubic root formulas as isometries of and its unit tangent bundle, respectively. Utilizing this geometry, we determine when the map sending certain families of polynomials to their complex roots (our starscape images) are embeddings. We reconsider the fundamental questions of the Diophantine approximation of complex numbers by algebraic numbers of bounded degree, from the geometric perspective developed. In the quadratic case (approximation by quadratic irrationals), we consider approximation in terms of hyperbolic distance between roots in the complex plane and the discriminant as a measure of arithmetic height on a polynomial. In particular, we determine the supremum on the exponent k for which an algebraic target α has infinitely many approximations β whose hyperbolic distance from α does not exceed . It turns out to fall into two cases, depending on whether α lies on the image of a plane of rational slope in coefficient space (a rational geodesic). The result comes as an application of Schmidt’s subspace theorem. Our results recover the quadratic case of results of Bugeaud and Evertse, and give some geometric explanation for the dichotomy they discovered. Our statements go a little further in distinguishing approximability in terms of whether the target or approximations lie on rational geodesics. The paper comes with accompanying software, and finishes with a wide variety of open problems.

Galois Distribution on Tori—A Refinement, Examples, and Applications

Abstract In this paper we give a refinement of the one-dimensional form of Bilu’s Equidistribution Theorem together with some examples and applications. These include an explicit bound for the sum of the $e^{i\theta }$ taken over all conjugates $re^{i\theta }$ of an algebraic number $\alpha $ in terms of its height $h$ and degree $d$. That is applied to show that a certain discrepancy of $\alpha $ is at most $8000h^{1/3}$, which removes an extra term involving $d$ from previous bounds; this cannot be done simply by using a lower bound for $h$ in terms of $d$, even assuming the Lehmer Conjecture. And we give a new estimate for the norm of $1-\alpha $. We also improve existing upper bounds for the height of $\xi $ when $\xi ,1-\xi $ are multiplicatively dependent.

On second order linear divisibility sequences over algebraic number fields

On second order linear divisibility sequences over algebraic number fields

Origamis associated to minimally intersecting filling pairs

Let $S_{g}$ denote the closed orientable surface of genus $g$. In joint work with Huang, the first author constructed exponentially-many (in $g$) mapping class group orbits of pairs of simple closed curves whose complement is a single topological disk. Using different techniques, we improve on this result by constructing factorially-many (again in $g$) such orbits. These new orbits are chosen so that the absolute value of the algebraic intersection number is equal to the geometric intersection number, implying that each pair naturally gives rise to an origami. We collect some rudimentary experimental data on the corresponding $SL(2, \mathbb{Z})$-orbits and suggest further study and conjectures.

Understanding mathematical abstraction in the formularization of Galileo's law

Galileo's revolution in science introduced an analytical method to science that typifies the overall modern thinking of extracting, abstracting, and grasping only critical aspects of the target phenomena and focusing on “how”, which is a quantitative relationship between variables, instead of “why”. For example, to him, the question of 'why does an object fall' is of no significance; instead, only the quantitative relationship between distance from the falling object and time is important. Yet, the most fundamental aspect of his idea is that he introduced a quantified time t. When an object is projected horizontally, the distance travelled at some time in the horizontal direction is summed up as d ∝t, whereas the distance falling at some time in the vertical direction is summed up as d ∝ t². Here, the distance, which is a spatial attribute, is expressed as a function of time, t. That is, time is identified as a homogeneous amount that can be reduced to an algebraic number. It is now possible to calculate the laws of motion of things using functions of time. In this respect, mathematical time was a decisive variable in making mathematisation of physical nature practical. Because, according to atomic theory, vacuum exists between an atom and an object composed of atoms or between objects – ignoring factors that interfere with motion, such as friction – the space for absolute time, which is a mathematical time, can be geometrically defined. In order to justify this mathematical abstraction strategy, thought experiments were conducted rather than laboratory experiments, which at that time were difficult to perform.

On a generalization of Menon–Sury identity to number fields involving a Dirichlet character

For every positive integer n, Sita Ramaiah’s identity states that $$\begin{aligned}&\sum _{a_1, a_2, a_1+a_2 \in (\mathbb {Z}/n\mathbb {Z})^*} \gcd (a_1+a_2-1,n) = \phi _2(n)\sigma _0(n) \\&\quad \text { where } \; \phi _2(n)= \sum _{a_1, a_2, a_1+a_2 \in (\mathbb {Z}/n\mathbb {Z})^*} 1, \end{aligned}$$where \((\mathbb {Z}/n\mathbb {Z})^*\) is the multiplicative group of units of the ring \(\mathbb {Z}/n\mathbb {Z}\) and \(\sigma _s(n) = \displaystyle \sum \nolimits _{d\mid n}d^s\). This identity can also be viewed as a generalization of Menon’s identity. In this article, we generalize this identity to an algebraic number field K involving a Dirichlet character \(\chi \). Our result is a further generalization of recent results in Ji and Wang (Ramanujan J 53:585–594, 2020) and Sury (Rend Circ Mat Palermo 58:99–108, 2009).

FPS in action

Linear recurrence equations with constant coefficients define the power series coefficients of rational functions. However, one usually prefers to have an explicit formula for the sequence of coefficients, provided that such a formula is "simple" enough. Simplicity is related to the compactness of the formula due to the presence of algebraic numbers: "the smaller, the simpler". This poster showcases the capacity of recent updates on the Formal Power Series (FPS) algorithm, implemented in Maxima and Maple (convert/FormalPowerSeries), to find simple formulas for sequences like those from https://oeis.org/A307717, https://oeis.org/A226782, or https://oeis.org/A226784 by computing power series representations of their correctly guessed generating functions. We designed the algorithm for the more general context of univariate P -recursive sequences. Our implementations are available at http://www.mathematik.uni-kassel.de/~bteguia/FPS_webpage/FPS.htm

On Finite Subgroups in the General Linear Groups over an Algebraic Number Field

As is well-known, there are only finitely many isomorphic classes of finite subgroups in a given general linear group over the field of rational numbers. This result can be generalized to any algebraic number field. While the case of field of rational numbers is relatively well-studied, we still do not know much for general algebraic number field cases. In this article, we discuss the finiteness of isomorphic classes of finite subgroups of general linear groups over an algebraic number field. We give a method to calculate a multiplicative bound for the orders of finite subgroups and to classify finite cyclic subgroups.

Jenks prize announcement

Pari/GP was designed for algebraic number theory. As such it does not have some standard features that other general purpose computer algebra systems have such as Gröbner bases and multivariate polynomial factorization. In this letter I say a few words about the history of Pari/GP, the impact Pari/GP has had on number theory, and some of the unique software design features of Pari/GP. I've also included a short bio from Henri, Bill and Karim.

Enumeration of rooted 3-connected bipartite planar maps

Abstract We provide the first solution to the problem of counting rooted 3-connected bipartite planar maps. Our starting point is the enumeration of bicoloured planar maps according to the number of edges and monochromatic edges, following Bernardi and Bousquet-Mélou [J. Comb. Theory Ser. B (2011)]. The decomposition of a map into 2- and 3-connected components allows us to obtain the generating functions of 2- and 3-connected bicoloured maps. Setting to zero the variable marking monochromatic edges we obtain the generating function of 3-connected bipartite maps, which is algebraic of degree 26. We deduce from it an asymptotic estimate for the number of 3-connected bipartite planar maps of the form t · n −5/2 γ n , where γ = ρ −1 ≈ 2.40958 and ρ ≈ 0.41501 is an algebraic number of degree 10.

Fractional parts of powers of real algebraic numbers

Let α be a real algebraic number greater than 1. We establish an effective lower bound for the distance between an integral power of α and its nearest integer.

The digit exchanges in the rotational beta expansions of algebraic numbers

In this article, we investigate the β-expansions of real algebraic numbers. In particular, we give new lower bounds for the number of digit exchanges in the case where β is a Pisot or Salem number. Moreover, we define a new class of algebraic numbers, quasi-Pisot numbers and quasi-Salem numbers, which gives a generalization of Pisot numbers and Salem numbers.Our method is applicable also to the digit expansions of complex algebraic numbers, which gives a new estimate. In particular, we investigate the digits of rotational beta expansion considered by Akiyama and Caalim [3] and zeta-expansion by Surer [20], where the base is a quasi-Pisot or quasi-Salem number.

Cyclotomic Construction of Sparse Code Multiple Access With Improved Diversity

A number of studies focus on sparse code multiple access (SCMA) to increase the spectral efficiency and support the massive connectivity required for future wireless communications networks. This work reports an improved design for SCMA based on algebraic number field theory and cyclotomic construction of a lattice. The proposed constellation improves the diversity of the SCMA system, which supports the probability of decoding error. We show that the proposed constellation in the uplink transmission of SCMA supports the spectral efficiency by 1 dB gain in SNR when the error probability of SCMA is 0.001 as compared to the proposals in the literature.

Northcott numbers for the house and the Weil height

For an algebraic number α $\alpha$ and γ ∈ R $\gamma \in \mathbb {R}$ , let be the house, h ( α ) $h(\alpha )$ be the (logarithmic) Weil height, and h γ ( α ) = ( deg α ) γ h ( α ) $h_\gamma (\alpha )=(\deg \alpha )^\gamma h(\alpha )$ be the γ $\gamma$ -weighted (logarithmic) Weil height of α $\alpha$ . Let f : Q ¯ → [ 0 , ∞ ) $f:\overline{\mathbb {Q}}\rightarrow [0,\infty )$ be a function on the algebraic numbers Q ¯ $\overline{\mathbb {Q}}$ , and let S ⊂ Q ¯ $S\subset \overline{\mathbb {Q}}$ . The Northcott number N f ( S ) $\mathcal {N}_f(S)$ of S $S$ , with respect to f $f$ , is the infimum of all X ⩾ 0 $X\geqslant 0$ such that { α ∈ S ; f ( α ) < X } $\lbrace \alpha \in S; f(\alpha )< X\rbrace$ is infinite. This paper studies the set of Northcott numbers N f ( O ) $\mathcal {N}_f({\mathcal {O}})$ for subrings of Q ¯ $\overline{\mathbb {Q}}$ for the house, the Weil height, and the γ $\gamma$ -weighted Weil height. We show: (1) Every t ⩾ 1 $t\geqslant 1$ is the Northcott number of a ring of integers of a field w.r.t. the house . (2) For each t ⩾ 0 $t\geqslant 0$ , there exists a field with Northcott number in [ t , 2 t ] $ [t,2t]$ w.r.t. the Weil height h ( · ) $h(\cdot )$ . (3) For all 0 ⩽ γ ⩽ 1 $0\leqslant \gamma \leqslant 1$ and γ ′ < γ $\gamma ^{\prime }<\gamma$ , there exists a field K $K$ with N h γ ′ ( K ) = 0 $\mathcal {N}_{h_{\gamma ^{\prime }}}(K)=0$ and N h γ ( K ) = ∞ $\mathcal {N}_{h_\gamma }(K)=\infty$ . For (1) we provide examples that satisfy an analogue of Julia Robinson's property (JR), examples that satisfy an analogue of Vidaux and Videla's isolation property, and examples that satisfy neither of those. Item (2) concerns a question raised by Vidaux and Videla due to its direct link with decidability theory via the Julia Robinson number. Item (3) is a strong generalisation of the known fact that there are fields that satisfy the Lehmer conjecture but which are not Bogomolov in the sense of Bombieri and Zannier.

Some Lacunary Power Series and Mahler’s Um-numbers in p-adic Domain

In this paper, we consider some lacunar power series in Qp, where p is a prime number. We obtain some results about lacunary power series with rational coefficients for Um-number arguments in Qp. We determined whether the values of the considered lacunar power series belong to a certain field of algebraic numbers or to the set of transcendental numbers in Qp. Also, we find similar results for lacunary power series with p-adic algebraic coefficients.

Torsion points on isogenous abelian varieties

Investigating a conjecture of Zannier, we study irreducible subvarieties of abelian schemes that dominate the base and contain a Zariski dense set of torsion points that lie on pairwise isogenous fibers. If everything is defined over the algebraic numbers and the abelian scheme has maximal variation, we prove that the geometric generic fiber of such a subvariety is a union of torsion cosets. We go on to prove fully or partially explicit versions of this result in fibered powers of the Legendre family of elliptic curves. Finally, we apply a recent result of Galateau and Martínez to obtain uniform bounds on the number of maximal torsion cosets in the Manin–Mumford problem across a given isogeny class. For the proofs, we adapt the strategy, due to Lang, Serre, Tate, and Hindry, of using Galois automorphisms that act on the torsion as homotheties to the family setting.