Extensions of local fields and truncated power series

Let K be a finite tamely ramified extension of Q_p and let L/K be a totally ramified (Z/pⁿZ)-extension. Let π_L be a uniformizer for L, let σ be a generator for Gal(L/K), and let f(X) be an element of O_K[X] such that σ(π_L)=f(π_L). We show that the reduction of f(X) modulo the maximal ideal of O_K determines a certain subextension of L/K up to isomorphism. We use this result to study the field extensions generated by periodic points of a p-adic dynamical system.     

Finite polynomial orbits in finitely generated domains

LetR be a commutative domain of zero characteristic and letf(X) be a polynomial with coefficients inR. It is shown that all finite orbits inR. under the mapping induced byR have their cardinalities bounded by a constant depending onR but not onf. In the case whenR is the ring of all integers in an algebraic number field this constant is effectively determined.     

Multiplication of a Schubert polynomial by a Schur polynomial

Schur polynomials are a special case of Schubert polynomials. In this paper, we give an algorithm to compute the product of a Schubert polynomial with a Schur polynomial on the basis of Schubert polynomials. This is a special case of the general problem of the multiplication of two Schubert polynomials, where the corresponding algorithm is still missing. The main tools for the given algorithm is a factorization property of a special class of Schubert polynomials and the transition formula for Schubert polynomials.     

Extensions of abelian varieties defined over a number field

We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with congruences between modular forms and special values of L-functions.     

Numerical factorization of a polynomial by rational Hermite interpolation

We derive a class of iterative formulae to find numerically a factor of arbitrary degree of a polynomialf(x) based on the rational Hermite interpolation. The iterative formula generates the sequence of polynomials which converge to a factor off(x). It has a high convergence order even for a factor which includes multiple zeros. Some numerical examples are also included.     

Enumeration of isomorphism classes of extensions of p-adic fields

Let Ω be an algebraic closure of and let F be a finite extension of contained in Ω. Given positive integers f and e, the number of extensions K/F contained in Ω with residue degree f and ramification index e was computed by Krasner. This paper is concerned with the number of F-isomorphism classes of such extensions. We determine completely when p2?e and get partial results when p2||e. When s is large, is equal to the number of isomorphism classes of finite commutative chain rings with residue field , ramification index e, and length s.     

Arithmetic height functions over finitely generated fields

In this paper, we propose a new height function for a variety defined over a finitely generated field over ℚ. For this height function, we prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (Manin-Mumford’s conjecture). Oblatum 7-VI-1999 & 21-IX-1999 / Published online: 24 January 2000     

Sharp bounds for the number of roots of univariate fewnomials

Let K be a field and t?0. Denote by Bm(t,K) the supremum of the number of roots in K^?, counted with multiplicities, that can have a non-zero polynomial in K[x] with at most t+1 monomial terms. We prove, using an unified approach based on Vandermonde determinants, that Bm(t,L)?t²Bm(t,K) for any local field L with a non-archimedean valuation v:L→R∪{∞} such that vZ≠0_|≡0 and residue field K, and that Bm(t,K)?(t²−t+1)(pf−1) for any finite extension K/Qp with residual class degree f and ramification index e, assuming that p>t+e. For any finite extension K/Qp, for p odd, we also show the lower bound Bm(t,K)?(2t−1)(pf−1), which gives the sharp estimation Bm(2,K)=3(pf−1) for trinomials when p>2+e.     

Global solvability of real analytic complex vector fields in two variables

This paper deals with the global solvability of a complex vector field with real analytic coefficients in two real variables. The vector field is assumed to satisfy the Nirenberg-Treves condition (P) for local solvability. Normal forms for the vector field near the one-dimensional orbits are obtained and a generalization of the Riemann-Hilbert problem is considered.     

On Opial-type inequalities in two variables

Summary In this paper we establish some new Opial-type inequalities in two variables which have a wide range of applications in the study of differential and integral equations.     

Images of locally finite derivations of polynomial algebras in two variables

In this paper we show that the image of any locally finite k-derivation of the polynomial algebra k[x,y] in two variables over a field k of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian conjecture is equivalent to the statement that the image of every k-derivation D of k[x,y] such that and is a Mathieu subspace of k[x,y].     

Abelian varieties over finitely generated fields and the conjecture of Geyer and Jarden on torsion

In this paper we prove the Geyer‐Jarden conjecture on the torsion part of the Mordell‐Weil group for a large class of abelian varieties defined over finitely generated fields of arbitrary characteristic. The class consists of all abelian varieties with big monodromy, i.e., such that the image of Galois representation on ?‐torsion points, for almost all primes ?, contains the full symplectic group.     

Two-dimensional representations of finitely generated free group over an arbitrary field

Working over an arbitrary field, we give equivalent conditions for a representation of a finitely generated free group with given traces and determinants to exist and to be reducible, respectively; also, we classify all two-dimensional representations of a finitely generated free group.     

On primitive roots of tori: The case of function fields

We generalize Bilharz's Theorem for to all one-dimensional tori over global function fields of finite constant field. As an application, we also derive an analogue, in the setting of function fields, of a theorem (Chen-Kitaoka-Yu, Roskam) on the distribution of fundamental units modulo primes. Received: 16 October 2000 / Published online: 2 December 2002 Research partially supported by National Science Council, Rep. of China.     

The computation of multiple roots of a polynomial

This paper considers structured matrix methods for the calculation of the theoretically exact roots of a polynomial whose coefficients are corrupted by noise, and whose exact form contains multiple roots. The addition of noise to the exact coefficients causes the multiple roots of the exact form of the polynomial to break up into simple roots, but the algorithms presented in this paper preserve the multiplicities of the roots. In particular, even though the given polynomial is corrupted by noise, and all computations are performed on these inexact coefficients, the algorithms ‘sew’ together the simple roots that originate from the same multiple root, thereby preserving the multiplicities of the roots of the theoretically exact form of the polynomial. The algorithms described in this paper do not require that the noise level imposed on the coefficients be known, and all parameters are calculated from the given inexact coefficients. Examples that demonstrate the theory are presented.     

Bounds on the chromatic polynomial and on the number of acyclic orientations of a graph

An upper bound is given on the number of acyclic orientations of a graph, in terms of the spectrum of its Laplacian. It is shown that this improves upon the previously known bound, which depended on the degree sequence of the graph. Estimates on the new bound are provided.A lower bound on the number of acyclic orientations of a graph is given, with the help of the probabilistic method. This argument can take advantage of structural properties of the graph: it is shown how to obtain stronger bounds for small-degree graphs of girth at least five, than are possible for arbitrary graphs. A simpler proof of the known lower bound for arbitrary graphs is also obtained.Both the upper and lower bounds are shown to extend to the general problem of bounding the chromatic polynomial from above and below along the negative real axis.Partially supported by the NSF under grant CCR-9404113. Most of this research was done while the author was at the Massachusetts Institute of Technology, and was supported by the Defense Advanced Research Projects Agency under Contracts N00014-92-J-1799 and N00014-91-J-1698, the Air Force under Contract F49620-92-J-0125, and the Army under Contract DAAL-03-86-K-0171.Supported by an ONR graduate fellowship, grants NSF 8912586 CCR and AFOSR 89-0271, and an NSF postdoctoral fellowship.     

On the existence of two non-neighboring subgraphs in a graph

Does there exist a functionf(r, n) such that each graphG with _Z (G)≧f(r, n) contains either a complete subgraph of orderr or else two non-neighboringn-chromatic subgraphs? It is known thatf(r, 2) exists and we establish the existence off(r, 3). We also give some interesting results about graphs which do not contain two independent edges.     

Maximum set of edges no two covered by a clique

Leth(G) be the largest number of edges of the graphG. no two of which are contained in the same clique. ForG without isolated vertices it is proved that ifh(G)≦5, thenχ( )≦h(G), but ifh(G)=6 thenχ( ) can be arbitrarily large.