Computing all power integral bases in orders of totally real cyclic sextic number fields

An algorithm is given for determining all power integral bases in orders of totally real cyclic sextic number fields. The orders considered are in most cases the maximal orders of the fields. The corresponding index form equation is reduced to a relative Thue equation of degree 3 over the quadratic subfield and to some inhomogeneous Thue equations of degree 3 over the rationals. At the end of the paper, numerical examples are given.

     

Power integral bases in a parametric family of totally real cyclic quintics

We consider the totally real cyclic quintic fields , generated by a root of the polynomial

Assuming that is square free, we compute explicitly an integral basis and a set of fundamental units of and prove that has a power integral basis only for . For (both values presenting the same field) all generators of power integral bases are computed.

     

Congruences for the class numbers of real cyclic sextic number fields

LetK ₆ be a real cyclic sextic number field, andK ₂,K ₃ its quadratic and cubic subfield. Leth(L) denote the ideal class number of fieldL. Seven congruences forh ^- =h (K ₆)/(h(K ₂)h(K ₃)) are obtained. In particular, when the conductorf ₆ ofK ₆ is a primep, , whereC is an explicitly given constant, andB _n is the Bernoulli number. These results on real cyclic sextic fields are an extension of the results on quadratic and cyclic quartic fields. Project supported by the National Natural Science Foundation of China (Grant No. 19771052).     

Relative integral bases over a Hilbert class field

Let K be a number field, H its Hilbert class field and L a Galois extension of K containing H. In this paper, we prove that L|H has a relative integral basis (RIB) if the order of G=Gal(L|H) is odd or if the 2-Sylow subgroups of G are not cyclic. If the order of G is even and the 2-Sylow subgroups are cyclic we reduce the problem of the existence of a RIB to a quadratic extension of H.     

Nonlinear stochastic integration with a non-smooth family of integrators

We consider nonlinear stochastic integrals of Itô-type w.r.t. a family of semimartingales which depend on a spatial parameter. These integrals were introduced by Carmona/Nualart, Kunita, and Le Jan. The extension of the elementary nonlinear integral is based on the condition that the semimartingale kernel has nice continuity properties in the spatial parameter. We investigate the case that continuity is not available and suggest different directions of generalization. This brings us beyond the case that any integral can be approximated by integrals with integrands taking only finitely many values.     

Exceptional units in a family of quartic number fields

We determine all exceptional units among the elements of certain groups of units in quartic number fields. These groups arise from a one-parameter family of polynomials with two real roots.

     

Fundamental units in a family of cyclic cubic fields

For a family of cyclic cubic fields, we give a system of fundamental units, we obtain conditions sufficient to force the evenness of the classnumber h_K, and we prove asymptotic results for the classnumbers of those fields.     

Générateurs de l'anneau des entiers d'une extension cyclotomique

Let p be an odd prime and q=pm, where m is a positive integer. Let ζq be a qth primitive root of 1 and Oq be the ring of integers in Q(ζq). In [I. Gaál, L. Robertson, Power integral bases in prime-power cyclotomic fields, J. Number Theory 120 (2006) 372-384] I. Gaál and L. Robertson show that if , where is the class number of , then if α∈Oq is a generator of Oq (in other words Z[α]=Oq) either α is equals to a conjugate of an integer translate of ζq or is an odd integer. In this paper we show that we can remove the hypothesis over . In other words we show that if α∈Oq is a generator of Oq then either α is a conjugate of an integer translate of ζq or is an odd integer.     

Approximation of potential integral by radial bases for solutions of Helmholtz equation

For a Helmholtz equation Δu(x) + κ ² u(x) = f(x) in a region of R ^s , s ≥ 2, where Δ is the Laplace operator and κ = a + ib is a complex number with b ≥ 0, a particular solution is given by a potential integral. In this paper the potential integral is approximated by using radial bases with the order of approximation derived.      

A rational sextic associated with a Desargues configuration

In this paper we consider a Desargues configuration in the projective plane, i.e. ten points and ten lines, on each line we have three of the points and through each point we have three of the lines. We construct a rational curve of order 6 which has a node at each of the ten points. We have never seen this kind of curve in the literature, but it is well known that for anyn there exists a rational curve of ordern which has [(n–1)(n–2)]/2 nodes and ifn=6 we find a sextic with ten nodes. The purpose of this paper is to obtain a sextic of this kind as a locus of points in connection with special projectivities of the plane associated with the Desargues configuration and to find a rational parametric representation of it. A large part of this paper is done with MACSYMA: it is an application of computer algebra in algebraic geometry. Special cases, where we find a quintic, a quartic or a cubic, are given in the last section.     

The growth of valuations on rational function fields in two variables

Given a valuation on the function field , we examine the set of images of nonzero elements of the underlying polynomial ring under this valuation. For an arbitrary field , a Noetherian power series is a map that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on . Although the value groups corresponding to such valuations are well-understood, the restrictions of the valuations to underlying polynomial rings have yet to be characterized. Let denote the images under the valuation of all nonzero polynomials of at most degree in the variable . We construct a bound for the growth of with respect to for arbitrary valuations, and then specialize to valuations that arise from Noetherian power series. We provide a sufficient condition for this bound to be tight.

     

On a family of symmetric polynomials

In this paper, we provide two simple approaches to the explicit expression of a family of symmetric polynomials introduced and studied in Milovanovi? and Cvetkovi? [J. Math. Anal. Appl. 311 (2005) 191], thereby improving on their observations.     

Subquadratic-time factoring of polynomials over finite fields

New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree over a finite field of constant cardinality in time . Previous algorithms required time . The new algorithms rely on fast matrix multiplication techniques. More generally, to factor a polynomial of degree over the finite field with elements, the algorithms use arithmetic operations in .

The new ``baby step/giant step' techniques used in our algorithms also yield new fast practical algorithms at super-quadratic asymptotic running time, and subquadratic-time methods for manipulating normal bases of finite fields.

     

A family of orthonormal wavelet bases with dilation factor 4

In this paper, we study a method for the construction of orthonormal wavelet bases with dilation factor 4. More precisely, for any integer M>0, we construct an orthonormal scaling filter mM(ξ) that generates a mother scaling function ?M, associated with the dilation factor 4. The computation of the different coefficients of ²|mM(ξ)| is done by the use of a simple iterative method. Also, this work shows how this construction method provides us with a whole family of compactly supported orthonormal wavelet bases with arbitrary high regularity. A first estimate of α(M), the asymptotic regularity of ?M is given by α(M)∼0.25M. Examples are provided to illustrate the results of this work.