On Σ-subsets of naturals over abelian groups

We obtain conditions for the Σ-definability of a subset of the set of naturals in the hereditarily finite admissible set over a model and for the computability of a family of such subsets. We prove that: for each e-ideal I there exists a torsion-free abelian group A such that the family of e-degrees of Σ-subsets of ω in $\mathbb{H}\mathbb{F}(A)$ coincides with I; there exists a completely reducible torsion-free abelian group in the hereditarily finite admissible set over which there exists no universal Σ-function; for each principal e-ideal I there exists a periodic abelian group A such that the family of e-degrees of Σ-subsets of ω in $\mathbb{H}\mathbb{F}(A)$ coincides with I.     

On an entropy of ℤ + k -actions

In this paper,a definition of entropy for Z＋k（k≥2）-actions due to Friedland is studied.Unlike the traditional definition,it may take a nonzero value for actions whose generators have finite（even zero） entropy as single transformations.Some basic properties are investigated and its value for the Z＋k-actions on circles generated by expanding endomorphisms is given.Moreover,an upper bound of this entropy for the Z＋k-actions on tori generated by expanding endomorphisms is obtained via the preimage entropies,which are entropy-like invariants depending on the ＂inverse orbits＂ structure of the system.     

On the computation of the Galois group over the quotient field of ℂ[λ]

Summary We give an algorithm for the computation of the Galois group of the splitting field of polynomials in two variables with integer coefficients over the quotient field (), (the rational functions in ). The algorithm uses a constructive version of the Newton polygon method and analytic continuations.Supported in part by the Fonds National Suisse     

On Mordell–Weil groups of isotrivial abelian varieties over function fields

We show that the Mordell–Weil rank of an isotrivial abelian variety with cyclic holonomy depends only on the fundamental group of the complement to the discriminant, provided the discriminant has singularities in CM class introduced here. This class of singularities includes all unibranched plane curves singularities. As a corollary, we describe a family of simple Jacobians over the field of rational functions in two variables for which the Mordell–Weil rank is arbitrarily large.     

On monochromatic solutions of some nonlinear equations in ℤ/pℤ

Let the set of positive integers be colored in an arbitrary way in finitely many colors (a “finite coloring”). Is it true that, in this case, there are x, y ∈ ? such that x + y, xy, and x have the same color? This well-known problem of the Ramsey theory is still unsolved. In the present paper, we answer this question in the affirmative in the group ?/p?, where p is a prime, and obtain an even stronger density result.     

On Selmer groups of abelian varieties over ℓ-adic Lie extensions of global function fields

Let F be a global function field of characteristic p > 0 and A/F an abelian variety. Let K/F be an ?-adic Lie extension (? ≠ p) unramified outside a finite set of primes S and such that Gal(K/F) has no elements of order ?. We shall prove that, under certain conditions, Sel _A (K) _? ^∨ has no nontrivial pseudo-null submodule.     

On ℤp-graded identities and cocharacters of the Grassmann algebra

Let p be an odd prime number, F a field of characteristic zero, and let Ebe the unitary Grassmann algebra generated by the infinite-dimensionalF-vector space L. We determine the bases of the ?_p-graded identities.Moreover we compute the ?_p-graded codimension and cocharacter sequences for the algebra E endowed with any ?_p-grading such that L is a homogeneous subspace.     

Finitary and Artinian-Finitary Groups over the Integers ℤ

In a series of papers, we have considered finitary (that is, Noetherian-finitary) and Artinian-finitary groups of automorphisms of arbitrary modules over arbitrary rings. The structural conclusions for these two classes of groups are really very similar, especially over commutative rings. The question arises of the extent to which each class is a subclass of the other.Here we resolve this question by concentrating just on the ground ring of the integers . We show that even over neither of these two classes of groups is contained in the other. On the other hand, we show how each group in either class can be built out of groups in the other class. This latter fact helps to explain the structural similarity of the groups in the two classes.     

On the Erd?s-Ginzburg-Ziv constant of finite abelian groups of high rank

Let G be a finite abelian group. The Erd?s-Ginzburg-Ziv constant s(G) of G is defined as the smallest integer l∈N such that every sequence S over G of length |S|?l has a zero-sum subsequence T of length |T|=exp(G). If G has rank at most two, then the precise value of s(G) is known (for cyclic groups this is the theorem of Erd?s-Ginzburg-Ziv). Only very little is known for groups of higher rank. In the present paper, we focus on groups of the form , with n,r∈N and n?2, and we tackle the study of s(G) with a new approach, combining the direct problem with the associated inverse problem.     

Abelian groups as UA-modules over the ring ℤ

Let V be a module over a ring R. Themodule V is called a unique addition module (a UA-module) if there is no new addition on the set V without changing the action of R on V. In the paper, the UA-modules over the ring ? are found.     

On the construction of normal wildly ramified extensions over Q p , (p ≠ 2)

Let p be an odd prime number, and let Q _p be the field of rational p-adic numbers.The aim of this work is the determination of the standard form of an Eisenstein polynomial defining a normal wildly ramified extension of Q _p . We prove first the equivalence between normality and cyclicity, give some essential normality conditions for the general case (degree p ⁿ ), then we solve the problem completely for the case (degree p ²) also, we obtain that the normality depends on seven congruences modulo p ^m between the coefficients of the considered polynomial with just m = 2 or 3. Note that the case (degree p) was solved by Öystein Ore (see Math. Annalen 102 (1930), 283–304). Also examples are given.     

On the semistability of certain Lazarsfeld–Mukai bundles on abelian surfaces

Let \(X=\mathscr {J}(\widetilde{\mathscr {C}})\), the Jacobian of a genus 2 curve \(\widetilde{\mathscr {C}}\) over \({\mathbb {C}}\), and let Y be the associated Kummer surface. Consider an ample line bundle \(L=\mathscr {O}(m\widetilde{\mathscr {C}})\) on X for an even number m, and its descent to Y, say \(L'\). We show that any dominating component of \({\mathscr {W}}^1_{d}(|L'|)\) corresponds to \(\mu _{L'}\)-stable Lazarsfeld–Mukai bundles on Y. Further, for a smooth curve \(C\in |L|\) and a base-point free \(g^1_d\) on C, say (A, V), we study the \(\mu _L\)-semistability of the rank-2 Lazarsfeld–Mukai bundle associated to (C, (A, V)) on X. Under certain assumptions on C and the \(g^1_d\), we show that the above Lazarsfeld–Mukai bundles are \(\mu _L\)-semistable.     

On the torsion in K_2 of a field

For a field F,let Gn(F) = {{a,Φn(a)} ∈ K2(F) | a,Φn(a) ∈ F*},where Φn(x) is the n-th cyclotomic polynomial.At first,by using Faltings' theorem on Mordell conjecture it is proved that if F is a number field and if n = 4,8,12 is a positive integer having a square factor then Gn(F) is not a subgroup of K2(F),and then by using the results of Manin,Grauert,Samuel and Li on Mordell conjecture theorem for function fields,a similar result is established for function fields over an algebraically closed field.     

Sum of Euler–Kronecker constants over consecutive cyclotomic fields

We give an asymptotic expansion of the sum of Euler–Kronecker constants related to cyclotomic fields with consecutive parameter.