Arithmetic Characterization Research Articles

On Abel’s Problem and Gauss Congruences

Abstract A classical problem due to Abel is to determine if a differential equation $y^{\prime}=\eta y$ admits a non-trivial solution $y$ algebraic over $\mathbb C(x)$ when $\eta $ is a given algebraic function over $\mathbb C(x)$. Risch designed an algorithm that, given $\eta $, determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when $\eta $ admits a Puiseux expansion with rational coefficients at some point in $\mathbb C\cup \{\infty \}$, which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of $y^{\prime}=\eta y$ if and only if the coefficients of the Puiseux expansion of $x\eta (x)$ at $0$ satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations $y^{\prime}=\eta y$ with an algebraic solution when $x\eta (x)$ is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present two other applications, namely to diagonals of rational fractions and to directed two-dimensional walks.

Characterization and generation of straight line segments on triangular cell grid

In this paper we are considering straight lines and straight line segments defined by two triangle centroids in the triangular cell grid. A generation algorithm determining all the triangles (triangular cells) that are crossed by a straight line segment is proposed. There is the particular case where straight lines or straight line segments cross a vertex of the grid. We propose an arithmetical characterization of such lines and line segments.

Pure Weierstrass gaps from a quotient of the Hermitian curve

In this paper, by employing some results on Kummer extensions, we give an arithmetic characterization of pure Weierstrass gaps at many totally ramified places on a quotient of the Hermitian curve, including the well-studied Hermitian curve as a special case. The cardinality of these pure gaps is explicitly investigated. In particular, the numbers of gaps and pure gaps at a pair of distinct places are determined precisely, which can be regarded as an extension of the previous work by Matthews (2001) considered Hermitian curves. Additionally, some concrete examples are provided to illustrate our results.

Jamison sequences in countably infinite discrete Abelian groups

We extend the definition of Jamison sequences in the context of topological abelian groups. Then we study such sequences when the abelian group is discrete and countably infinite. An arithmetical characterization of such sequences is obtained, extending the result of Badea and Grivaux about Jamison sequences of integers. In particular, we prove that the sequence consisting of all elements of the group is a Jamison sequence. In the opposite, a sequence which generates a subgroup of infinite index in the group is never a Jamison sequence.

Bounded type interval exchange maps

Irrational numbers of bounded type have several equivalent characterizations. They have bounded partial quotients in terms of arithmetic characterization and in the dynamics of the circle rotation, the rescaled recurrence time to r-ball of the initial point is bounded below. In this paper, we consider how the bounded type condition of irrational is generalized into interval exchange maps.

On Rédei’s dihedral extension and triple reciprocity law

In this paper, we give an arithmetic characterization of Redei’s dihedral extension over $\mathbf{Q}$ and another simple proof of the reciprocity law of the triple symbol.

Products of two atoms in Krull monoids and arithmetical characterizations of class groups

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor and let D(G) be the Davenport constant of G. Then a product of two atoms of H can be written as a product of at most D(G) atoms. We study this extremal case and consider the set U{2,D(G)}(H) defined as the set of all l∈N with the following property: there are two atoms u,v∈H such that uv can be written as a product of l atoms as well as a product of D(G) atoms. If G is cyclic, then U{2,D(G)}(H)={2,D(G)}. If G has rank two, then we show that (apart from some exceptional cases) U{2,D(G)}(H)=[2,D(G)]∖{3}. This result is based on the recent characterization of all minimal zero-sum sequences of maximal length over groups of rank two. As a consequence, we are able to show that the arithmetical factorization properties encoded in the sets of lengths of a rank 2 prime power order group uniquely characterizes the group.

Diophantine type of interval exchange maps

AbstractRoth type irrational rotation numbers have several equivalent arithmetical characterizations as well as several equivalent characterizations in terms of the dynamics of the corresponding circle rotations. In this paper we investigate how to generalize Roth-like Diophantine conditions to interval exchange maps. If one considers the dynamics in parameter space one can introduce two non-equivalent Roth type conditions, the first (condition (Z)) by means of the Zorich cocycle [Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Ann. Inst. Fourier 46(2) (1996), 325–370], the second (condition (A)) by means of a further acceleration of the continued fraction algorithm by Marmi–Moussa–Yoccoz introduced in [The cohomological equation for Roth type interval exchange maps, J. Amer. Math. Soc. 18 (2005), 823–872]. A third very natural condition (condition (D)) arises by considering the distance between the discontinuity points of the iterates of the map. If one considers the dynamics of an interval exchange map in phase space then one can introduce the notion of Diophantine type by considering the asymptotic scaling of return times pointwise or with respect to uniform convergence (respectively conditions (R) and (U)). In the case of circle rotations all the above conditions are equivalent. For interval exchange maps of three intervals we show that (D) and (A) are equivalent and imply (Z), (U) and (R), which are equivalent among them. For maps of four intervals or more we prove several results; the only relation that we cannot decide is whether (Z) implies (R) or not.

The Inverse Problem Associated to the Davenport Constant for $C_2\oplus C_2 \oplus C_{2n}$, and Applications to the Arithmetical Characterization of Class Groups

The inverse problem associated to the Davenport constant for some finite abelian group is the problem of determining the structure of all minimal zero-sum sequences of maximal length over this group, and more generally of long minimal zero-sum sequences. Results on the maximal multiplicity of an element in a long minimal zero-sum sequence for groups with large exponent are obtained. For groups of the form $C_2^{r-1}\oplus C_{2n}$ the results are optimal up to an absolute constant. And, the inverse problem, for sequences of maximal length, is solved completely for groups of the form $C_2^2 \oplus C_{2n}$. Some applications of this latter result are presented. In particular, a characterization, via the system of sets of lengths, of the class group of rings of algebraic integers is obtained for certain types of groups, including $C_2^2 \oplus C_{2n}$ and $C_3 \oplus C_{3n}$; and the Davenport constants of groups of the form $C_4^2 \oplus C_{4n}$ and $C_6^2 \oplus C_{6n}$ are determined.

Arithmetical characterization of class groups of the form ℤ/nℤ⊕ℤ/nℤ via the system of sets of lengths

Let H be a Krull monoid with finite class group such that each class contains a prime divisor (e.g., the multiplicative monoid of the ring of algebraic integers of some number field). It is shown that it can be determined whether the class group is of the form ℤ/nℤ⊕ℤ/nℤ, for n≥3, just by considering the system of sets of lengths of H.

Recognition by spectrum for finite simple groups of Lie type

The goal of this article is to survey new results on the recognition problem. We focus our attention on the methods recently developed in this area. In each section, we formulate related open problems. In the last two sections, we review arithmetical characterization of spectra of finite simple groups and conclude with a list of groups for which the recognition problem was solved within the last three years.

Automorphic functions for the Whitehead-link-complement group

We construct automorphic functions on the real $3$-dimensional hyperbolic space $\mathbb{H}^3$ for the Whitehead-link-complement group $W\subset \mathit{GL}_2(\mathbb{Z}[i])$ and for a few groups commensurable with $W$. These automorphic functions give embeddings of the orbit spaces of $\mathbb{H}^3$ under these groups, and arithmetical characterizations of them.

Topological mixing for substitutions on two letters

We investigate topological mixing for $\mathbb Z$ and $\mathbb R$ actions associated with primitive substitutions on two letters. The characterization is complete if the second eigenvalue $\theta_2$ of the substitution matrix satisfies $|\theta_2|\ne 1$. If $|\theta_2|<1$, then (as is well known) the substitution system is not topologically weak mixing, so it is not topologically mixing. We prove that if $|\theta_2|> 1$, then topological mixing is equivalent to topological weak mixing, which has an explicit arithmetic characterization. The case $|\theta_2|=1$ is more delicate, and we only obtain some partial results.

Arithmetic differential equations and $E$-functions

Let $K$ be a number field. We give an arithmetic characterization at infinity of the differential operator of $K[x,d/dx]$ with minimal degree in $x$ annihilating a given $E$-function.

Arithmetical definability and computational complexity

In this paper, we introduce and study some syntactical fragments of monadic second-order and first-order (P EANO) arithmetic which we will prove the connection to famous complexity classes. Starting from descriptive complexity results, and giving an effective method for translating formulas between different logical structures representing encodings of integers, we give some new arithmetical characterizations of NP, PH, NL, and P.

An Arithmetic Characterization of Decomposition Methods in Banach Spaces Similar to Pełczyński's Decomposition Method

Inspired by Pełczyński's decomposition method in Banach spaces, we introduce the notion of Schroeder&#8211;Bernstein quadruples for Banach spaces. Then we use some Banach spaces constructed by W. T. Gowers and B. Maurey in 1997 to characterize them.

On Buchsbaum simplicial affine semigroups

We give an arithmetic characterization which allow us to determine algorithmically when the semigroup ring associated to a simplicial affine semigroup is Buchsbaum. This characterization is based on a test performed on the Apery sets of the extremal rays of the semigroup. We use this method to obtain the cardinality of minimal presentations for semigroups with minimal Apery set.

Strengths and Weaknesses of LH Arithmetic

In this paper we provide a new arithmetic characterization of the levels of the og-time hierarchy (LH). We define arithmetic classes and that correspond to -LOGTIME and -LOGTIME, respectively. We break and into natural hierarchies of subclasses and . We then define bounded arithmetic deduction systems ′ whose -definable functions are precisely B(-LOGTIME). We show these theories are quite strong in that (1) LIOpen proves for any fixed m that , (2) TAC, a theory that is slightly stronger than ′ whose (LH)-definable functions are LH, proves LH is not equal to -TIME(s) for any m> 0, where 2s ∈L, s(n) ∈ ω(log n), and (3) TAC proves LH ≠ for all k and m. We then show that the theory TAC cannot prove the collapse of the polynomial hierarchy. Thus any such proof, if it exists, must be argued in a stronger systems than ours.

AN ARITHMETICAL CHARACTERIZATION OF FINITE ELEMENTARY 2-GROUPS

Beginning with Carlitz's well known characterization of algebraic number rings with class number two, arithmetical characterizations of divisor class groups have been a topic of interest in the literature. In this note we develop a characterization of Dedekind domains with finite elementary 2-group class group. The characterization is in terms of the asymptotic behavior of the number of distinct factorizations of powers of an irreducible element.

Endo-circulant digraphs of degree two and Cayley digraphs on finite abelian groups

We extend the characterization of the c-circulant digraphs of degree two which are circulant to the endo-circulant digraphs. As a consequence, an arithmetic characterization of the endo-circulant digraphs which are circulant is obtained. Moreover, a theorem by Delorme et al. describing the Ádám isomorphisms for Cayley digraphs on finite abelian groups is seen from the point of view of isomorphic endo-circulant digraphs.