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It was proved by Roth in a recent paper that if α is any real algebraic number, and if K > 2, then the inequalityhas only a finite number of solutions in relatively prime integers p, q (q > 0) The object of the present paper is to prove that the lower bound for κ can be reduced if conditions are imposed on p and q. The result obtained is as follows.

Let F be a field, let ω ∈ F, and let n ⩾ 2 be a natural number. In this paper we define the ω-commuting graph of Mn(F), denoted by Γω(Mn(F)) which is a directed graph. We prove some theorems about the strong connectivity of this graph. Also we show that the induced directed subgraphs on the set of all reducible matrices and the set of all triangularizable matrices are strongly connected graphs. Among other results we determine the diameters of the induced directed subgraphs on the sets of all non-invertible and nilpotent matrices in Mn(F), exactly. Finally, we find good upper bounds for the diameters of some induced directed subgraphs of Γω(Mn(F)), such as the induced directed subgraphs on the set of all idempotent and diagonazable matrices in Mn(F). © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Buchi's problem asked whether there exists an integer $M$ such that the surface defined by a system of equations of the form $$x_{n}^2+x_{n-2}^2=2x_{n-1}^2+2,\quad n=2,\dotsc, M-1,$$ has no integer points other than those that satisfy $\pm x_n=\pm x_0+n$ (the $\pm$ signs are independent). If answered positively, it would imply that there is no algorithm which decides, given an arbitrary system $Q=(q_1,\dotsc,q_r)$ of integral quadratic forms and an arbitrary $r$-tuple $B=(b_1,\dotsc,b_r)$ of integers, whether $Q$ represents $B$ (see T. Pheidas and X. Vidaux, Fund. Math. 185 (2005) 171-194). Thus it would imply the following strengthening of the negative answer to Hilbert's tenth problem: the positive-existential theory of the rational integers in the language of addition and a predicate for the property '$x$ is a square' would be undecidable. Despite some progress, including a conditional positive answer (depending on conjectures of Lang), Buchi's problem remains open. In this paper we prove the following: an analogue of Buchi's problem in rings of polynomials of characteristic either 0 or $p\geq17$ and for fields of rational functions of characteristic 0; and an analogue of Buchi's problem in fields of rational functions of characteristic $p\geq19$, but only for sequences that satisfy a certain additional hypothesis. As a consequence we prove the following result in logic. Let $F$ be a field of characteristic either 0 or at least 17 and let $t$ be a variable. Let $L_{t}$ be the first order language which contains symbols for 0 and 1, a symbol for addition, a symbol for the property '$x$ is a square' and symbols for multiplication by each element of the image of $\mathbb{Z}[t]$ in $F[t]$. Let $R$ be a subring of $F(t)$, containing the natural image of $\mathbb{Z}[t]$ in $F(t)$. Assume that one of the following is true: $R\subset F[t]$; the characteristic of $F$ is either 0 or $p\geq19$. Then multiplication is positive-existentially definable over the ring $R$, in the language $L_t$. Hence the positive-existential theory of $R$ in $L_{t}$ is decidable if and only if the positive-existential ring-theory of $R$ in the language of rings, augmented by a constant-symbol for $t$, is decidable

Let E_G be a family of hyperelliptic curves over F2^(alg cl) with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm to compute the zeta function of E_g for g in a degree n extension field of F, which has as time complexity O(n^3) and memory requirements O(n^2). With a slightly different algorithm we can get time O(n^2,667) and memory O(n^2,5), and the computation of O(n) curves of the family can be done in time and space O(n^3). All these algorithms are polynomial in the genus., Comment: 23 pages

In this article we show that the set of Dirichlet regular boundary points of a bounded domain of dimension up to 4, definable in an arbitrary o-minimal structure on the field ℝ, is definable in the same structure. Moreover we give estimates for the dimension of the set of non-regular boundary points, depending on whether the structure is polynomially bounded or not. This paper extends the results from the author's Ph.D. thesis [6, 7] where the problem was solved for polynomially bounded o-minimal structures expanding the real field. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The maximal subgroups of the finite classical groups are divided by a theorem of Aschbacher into nine classes. In this paper, the authors show how to construct those maximal subgroups of the finite classical groups of linear, symplectic or unitary type that lie in the first eight of these classes. The ninth class consists roughly of absolutely irreducible groups that are almost simple modulo scalars, other than classical groups over the same field in their natural representation. All of these constructions can be carried out in low-degree polynomial time.

In this paper we continue to study the spectral norms and their completions ([4]) in the case of the algebraic closure of ℚ in ℂ. Let be the completion of relative to the spectral norm. We prove that can be identified with the R-subalgebra of all symmetric functions of C(G), where C(G) denotes the ℂ-Banach algebra of all continuous functions defined on the absolute Galois group G = Gal. We prove that any compact, closed to conjugation subset of ℂ is the pseudo-orbit of a suitable element of . We also prove that the topological closure of any algebraic number field in is of the form with x in .

Throughout this paper, we shall suppose that all algebraic number fields, namely, all algebraic extensions over the rational field , are contained in the complex field . Let be the set of all prime numbers. For any algebraic number field , let denote the ideal class group of and, writing for the maximal real subfield of , let denote the kernel of the norm map from to the ideal class group of ; for each , let denote the -class group of , that is, the -primary component of , and let denote the -primary component of . Furthermore, for each , we denote by the ring of -adic integers.

In this paper, further insight is obtained into the earlier approach of studying residually transcendental extensions of a valuation v of a field K to a simple transcendental extension K(x) of K by means of minimal pairs, thereby introducing new invariants corresponding to any element of an algebraic closure λ of K. It is also shown that these invariants are of independent interest as well. A characterization of those elements a belonging to λ is given such that there exists a minimal pair (a, δ) for some δ in the divisible closure of the value group of v.

Let Z be the centre of the group algebra of a symmetric group [Sscr ](n) over a field F characteristic p. One of the principal results of this paper is that the image of the Frobenius map z → zp, for z ∈ Z, lies in span Zp′ of the p-regular class sums. When p = 2, the image even coincides with Z2′. Furthermore, in all cases Zp′ forms a subalgebra of Z. Let pt be the p-exponent of [Sscr ](n). Then jpt = 0, for each element j of the Jacobson radical J of Z. It is shown that there exists j ∈ J such that jpt−1 ≠ 0. Most of the results are formulated in terms of the p-blocks of [Sscr ](n).

In this paper we expand the theory of connected components in non-archimedean discrete dynamical systems. We define two types of components and discuss their uses and applications in the study of dynamics of a rational function defined over a non-archimedean field K. We prove that some fundamental conjectures, including the No Wandering Domains conjecture, are equivalent, regardless of which definition of 'component' is used. We derive several results on the geometry of our components and the existence of periodic points within them. We also give a number of examples of p-adic maps with interesting or pathological dynamics.2000 Mathematical Subject Classification: primary 37B99; secondary 11S99, 30D05.

The Manin–Mumford conjecture asserts that if K is a field of characteristic zero, C a smooth proper geometrically irreducible curve over K, and J the Jacobian of C, then for any embedding of C in J, the set C(K) ∩ J(K)tors is finite. Although the conjecture was proved by Raynaud in 1983, and several other proofs have appeared since, a number of natural questions remain open, notably concerning bounds on the size of the intersection and the complete determination of C(K) ∩ J(K)tors for special families of curves C. The first half of this survey paper presents the Manin–Mumford conjecture and related general results, while the second describes recent work mostly dealing with the above questions.

Let v be a henselian valuation of a field K. In this paper it is proved that any finite extension (K′, v′) of (K, v) is tame if and only if there exists α ≠ 0 in K′ such that v′(α) = v(Tr K′/K (α)) using elementary results of valuation theory. A special case of this result, when the characteristic of the residue field of v is ρ > 0 and (K′, v′)/(K, v) is an extension of degree ρ, was proved in 1990 by J. P. Tignol (J. Reine Angew. Math. 404 (1990) 1–38). 1991 Mathematics Subject Classification 12J10, 12J25, 13A18.

Let ν be a valuation of any rank of a field K with value group G ν and f(X)= X m + a l X m−1 + … + a m be a polynomial over K. In this paper, it is shown that if (ν(a i )/i)≥(ν(a m )/m)>0 for l≤i≤m, and there does not exist any integer r>1 dividing m such that ν(a m )/r∈G ν , then f(X) is irreducible over K. It is derived as a special case of a more general result proved here. It generalizes the usual Eisenstein Irreducibility Criterion and an Irreducibility Criterion due to Popescu and Zaharescu for discrete, rank-1 valued fields, (cf. [Journal of Number Theory, 52 (1995), 98–118]).

Let V 0 be a discrete real valuation of a field K and x an indeterminate. In 1936, MacLane [3] gave a method of constructing all real valuations of K(x) which are extensions of V 0 .In this paper, we determine explicitly all rank 2 valuations of K(x) which extend V 0 .One can thereby describe all rank 2 valuations of K(x, y) which are trivial on an arbitrary K; x, y being algebraically independent over the field K. The latter valuations have been considered by Zariski [5] in the case when K is an algebraically closed field of characteristic zero.

Let K be an algebraic extension field of Q. Further, let 𝔒K be the domain of algebraic integers of K, and let Σn(x) be the n-th cyclotomic polynomial. This paper is devoted to the factorization of the principal ideal Σn(e) 𝔒K(e𝔒K) into prime ideals of 𝔒K. The main result (Theorem 3.4) can be considered as a generalization of a known result of Sylvester, Kronecker and Zsigmondy on the prime factorization of Σn(e) (eϵZ). With Theorem 3.4., we improve corresponding results of Redei [11] and Sachs [14]. We generalize a technique developed in [8] and [3] and we study also the cases that K is a quadratic field and a cyclotomic field, respectively. Finally, we apply the results to the parameter determination of Fourier-like number-theoretic transforms in 𝔒K.

In this paper we are interested in two related measures of the degree of approximation of a complex number ζ by algebraic numbers. For a given integer n ≥ 1, write wn(ζ) for the supremum of the exponents w for whichfor infinitely many polynomialsin Z[x] of height H(P) = max |av|. Clearly 0 ≤ w1 (ζ) ≤ w2 (ζ) ≤ …. On the other hand, write for the supremum of the exponents w for whichfor infinitely many algebraic numbers α of degree at most n.

In 1934 Gelfond [2] and Schneider [6] proved, independently, that the logarithm of an algebraic number to an algebraic base, other than 0 or 1, is either rational or transcendental and thereby solved the famous seventh problem of Hilbert. Among the many subsequent developments (cf. [4, 7, 8]), Gelfond [3] obtained, by means of a refinement of the method of proof, a positive lower bound for the absolute value of β1 log α1+β2 log α2, where β1, β2 denote algebraic numbers, not both 0, and α1,α1, denote algebraic numbers not 0 or 1, with log α1/log α2 irrational. Of particular interest is the special case in which β1,β2 denote integers. In this case it is easy to obtain a trivial positive lower bound (cf. [1; Lemma 2]), and the existence of a non-trivial bound follows from the Thue–Siegel–Roth theorem (see [4; Ch. I]). But Gelfond's result improves substantially on the former, and, unlike the latter, it is derived by an effective method of proof. Gelfond [4; p. 177] remarked that an analogous theorem for linear forms in arbitrarily many logarithms of algebraic numbers would be of great value for the solution of some apparently very difficult problems of number theory. It is the object of this paper to establish such a result.

The object of this paper is first to generalize the basic inequality of the large sieve method to exponential sums in many variables, and then to deduce results for algebraic number fields that are analogous to known results for the rational field.