Field Of Power Series Research Articles

Inhomogeneous Diophantine approximation over fields of formal power series

We prove a sharp analogue of Minkowski's inhomogeneous approximation theorem over fields of power series $\mathbb {F}_q((T^{-1}))$. Furthermore, we study the approximation to a given point $\underline {y}$ in $\mathbb {F}_q((T^{-1}))^2$ by the $\mathrm {SL}_2(\mathbb {F}_q[T])$-orbit of a given point $\underline {x}$ in $\mathbb {F}_q((T^{-1}))^2$.

On the Problem of Periodicity of Continued Fraction Expansions of $$\sqrt f $$for Cubic Polynomials over Number Fields

We obtain a complete description of fields $$\mathbb{K}$$ that are quadratic extensions of $$\mathbb{Q}$$ and of cubic polynomials $$f \in \mathbb{K}[x]$$ for which a continued fraction expansion of $$\sqrt f $$ in the field of formal power series $$\mathbb{K}((x))$$ is periodic. We also prove a finiteness theorem for cubic polynomials $$f \in \mathbb{K}[x]$$ with a periodic expansion of $$\sqrt f $$ over cubic and quartic extensions of $$\mathbb{Q}$$ .

On the estimation of coefficients of irreducible factors of polynomials over a field of formal power series in nonzero characteristic

We discuss some problems and results related to the Newton-Puiseux algorithm and its generalization for nonzero characteristic obtained by the author earlier. A new method is suggested for obtaining effective estimations of the roots of a polynomial in the field of fraction-power series in arbitrary characteristic.

On homogeneous and inhomogeneous Diophantine approximation over the fields of formal power series

We prove over fields of power series the analogues of several Diophantine approximation results obtained over the field of real numbers. In particular we establish the power series analogue of Kronecker's theorem for matrices, together with a quantitative form of it, which can also be seen as a transference inequality between uniform approximation and inhomogeneous approximation. Special attention is devoted to the one dimensional case. Namely, we give a necessary and sufficient condition on an irrational power series $\alpha$ which ensures that, for some positive $\eps$, the set $$ \liminf_{Q \in \mathbb{F}_q[z], \,\, \deg Q \to \infty} \| Q \| \cdot |\langle Q \alpha - \theta \rangle| \geq \eps $$ has full Hausdorff dimension.

On the Estimation of Coefficients of Irreducible Factors of Polynomials over a Field of Formal Power Series in Nonzero Characteristic

We discuss some results and problems related to the Newton–Puiseux algorithm and its generalization for nonzero characteristic obtained by the author earlier. A new method is suggested for obtaining effective estimates of the roots of a polynomial in the field of fractional power series in the case of arbitrary characteristic.

On The Geometric Continued Fractions in Positive Characteristic

In this paper we study another form in the field of formal power series over a finite field. If the continued fraction of a formal power seriesin $\mathbb{F}_q((X^{-1}))$ begins with sufficiently largegeometric blocks, then $f$ is transcendental.

On the topological structure of a Hahn field and convergence of power series

In this paper, we study the topological structure of the Hahn field F whose elements are functions from the additive abelian group of rational numbers Q to the real numbers field R, with well-ordered support. After reviewing the algebraic and order structures of the field F, we introduce different vector topologies on F that are induced by families of semi-norms, all of which are weaker than the order or valuation topology. We compare those vector topologies and we identify the weakest one which we denote by τw and whose properties are similar to those of the weak topology on the Levi-Civita field (Shamseddine, 2010). In particular, we state and prove a convergence criterion for power series in F,τw that is similar to that for power series on the Levi-Civita field in its weak topology (Shamseddine, 2013).

Convergence of β-Modified Jacobi-Perron Algorithm over the Field of Formal Power Series

The aim of this paper is to study multidimentional $\beta$-continued fraction algorithm over the field of formal power series. In the case of the Modified Jacobi-Perron algorithm, we prove that it converges.

Nonarchimedean quadratic Lagrange spectra and continued fractions in power series fields

Let $\mathbb F_q$ be a finite field of order a positive power $q$ of a prime number. We study the nonarchimedean quadratic Lagrange spectrum defined by Parkkonen and Paulin by considering the approximation by elements of the orbit of a given quadratic pow

Irreducibility criterion and 2-pisot series in positive character

Let Fq((X?1)) be the field of formal power series over a finite field Fq. We characterize a pair of roots that lies outside the unit disc while all remaining conjugates have a modulus strictly less than 1. In particular, we provide a sufficient condition for a pair of formal power series to be a 2-Pisot series. We also give an irreducibility criterion over Fq [X].

Bounds on the lengths of certain series expansions

In the real number field, there are several unique series expansions for each A ∈ (0, 1). Of interest are the Sylvester and alternating Sylvester series expansions since both expansions are finite if and only if A is rational. We obtain upper bounds on the length of rational A ∈ (0, 1) and lower bound on the length of certain classes of rational numbers. In the power series fields, let denote the finite field of q elements, let p(x) be an irreducible polynomial in , and let , respectively, be the completions of with respect to the p(x)-adic valuation, respectively, the infinite valuation. It is known that each , respectively, , subject to a technical assumption, has a unique Oppenheim series expansion, and such expansion is finite if and only if . Upper bounds on the length of these series expansions for are also derived.

A CONSTRUCTION OF SURFACES WITH LARGE HIGHER CHOW GROUPS

In this paper, we construct surfaces in $\mathbf{P}^{3}$ with large higher Chow groups defined over a Laurent power series field. Explicit elements in higher Chow group are constructed using configurations of lines contained in the surfaces. To prove the independentness, we compute the extension class in the Galois cohomologies by comparing them with the classical monodromies. It is reduced to the computation of linear algebra using monodromy weight spectral sequences.

Support of Laurent series algebraic over the field of formal power series

This work is devoted to the study of the support of a Laurent series in several variables which is algebraic over the ring of power series over a characteristic zero field. Our first result is the existence of a kind of maximal dual cone of the support of such a Laurent series. As an application of this result we provide a gap theorem for Laurent series which are algebraic over the field of formal power series. We also relate these results to diophantine properties of the fields of Laurent series.

Baker–Schmidt theorem for Hausdorff dimensions in finite characteristic

In 1962 Sprindžuk proved Mahler's conjecture in both the real and complex cases (as well as for p-adic numbers and for fields of formal power series over finite fields). Baker gave a generalized result by using a modified version of Sprindžuk's method. Later, Baker and Schmidt derived the Hausdorff dimensions of sets which are defined in terms of approximation by algebraic numbers of bounded degrees by using Baker's theorem. In this article we will prove two analogue theorems in the fields of formal power series over finite fields.

On periodic Jacobi-Perron algorithm over the field of formal power series

On periodic Jacobi-Perron algorithm over the field of formal power series

Endomorphisms of power series fields and residue fields of Fargues-Fontaine curves

We show that for k k a perfect field of characteristic p p , there exist endomorphisms of the completed algebraic closure of k ( ( t ) ) k((t)) which are not bijective. As a corollary, we resolve a question of Fargues and Fontaine by showing that for p p a prime and C p \mathbb {C}_p a completed algebraic closure of Q p \mathbb {Q}_p , there exist closed points of the Fargues-Fontaine curve associated to C p \mathbb {C}_p whose residue fields are not (even abstractly) isomorphic to C p \mathbb {C}_p as topological fields.

On the periodicity of continued fractions in hyperelliptic fields

Let L be a function field of a hyperelliptic curve defined over an arbitrary field characteristic different from 2. We construct an arithmetic of continued fractions of an arbitrary quadratic irrationality in field of formal power series with respect to linear finite valuation. The set of infinite valuation and finite linear valuation of L is denoted by S. As an application, we have found a relationship between the issue of the existence of nontrivial S-units in L and periodicity of continued fractions of some key elements of L.

On a two-valued sequence and related continued fractions in power series fields

We explicitly describe a noteworthy transcendental continued fraction in the field of power series over Q, having irrationality measure equal to 3. This continued fraction is a generating function of a particular sequence in the set {1, 2}. The origin of this sequence, whose study was initiated in a recent paper, is to be found in another continued fraction, in the field of power series over $\mathbb{F}_3$, which satisfies a simple algebraic equation of degree 4, introduced thirty years ago by D. Robbins.

About the algebraic closure of the field of power series in several variables in characteristic zero

We begin this paper by constructing different algebraically closed fields containing an algebraic closure of the field of power series in several variables over a characteristic zero field. Each of these fields depends on the choice of an Abhyankar valuation and is constructed via a generalization of the Newton-Puiseux method for this valuation. Then we study the Galois group of a polynomial with power series coefficients. In particular by examining more carefully the case of monomial valuations we are able to give several results concerning the Galois group of a polynomial whose discriminant is a weighted homogeneous polynomial times a unit. One of our main results is a generalization of Abhyankar-Jung Theorem for such polynomials, classical Abhyankar-Jung Theorem being devoted to polynomials whose discriminant is a monomial times a unit.

S-units in hyperelliptic fields and periodicity of continued fractions

Given a polynomial f of odd degree, the nontrivial S-units can be effectively related to the continued fraction expansions of the elements associated with \(\sqrt f \) only in the case where S contains an infinite valuation and a finite valuation determined by first-degree polynomial. A quasi-periodicity criterion for any element of the field of formal power series in a first-degree polynomial is obtained. For key elements, a more accurate criterion is found. The criterion is used to show that, for S specified above, in the presence of a nontrivial S-unit, the expansion of \(\sqrt f \) can be both nonperiodic and periodic. Estimates relating the quasi-period to the degree of the fundamental S-unit are obtained. Examples in which the bounds of these estimates are attained are given.