Field Of Formal Power Series Research Articles

Bounding the Pythagoras number of a field by 2n + 1

Given a positive integer n, a sufficient condition on a field is given for bounding its Pythagoras number by 2n+1. The condition is satisfied for n=1 by function fields of curves over iterated formal power series fields over R, as well as by finite field extensions of R((t0,t1)). In both cases, one retrieves the upper bound 3 on the Pythagoras number. The new method presented here might help to establish more generally 2n+1 as an upper bound for the Pythagoras number of function fields of curves over R((t1,…,tn)) and for finite field extensions of R((t0,…,tn)).

Convergence speed of the Brun algorithm over formal power series field

Convergence speed of the Brun algorithm over formal power series field

ON THE FINITENESS OF THE SET OF GENERALIZED JACOBIANS WITH NONTRIVIAL TORSION POINTS OVER ALGEBRAIC NUMBER FIELDS

For a smooth projective curve \(\mathcal{C}\) defined over algebraic number field k, we investigate the question of finiteness of the set of generalized Jacobians \({{J}_{\mathfrak{m}}}\) of a curve \(\mathcal{C}\) associated with modules \(\mathfrak{m}\) defined over k such that a fixed divisor representing a class of finite order in the Jacobian J of the curve \(\mathcal{C}\) provides the torsion class in the generalized Jacobian \({{J}_{\mathfrak{m}}}\). Various results on the finiteness and infiniteness of the set of generalized Jacobians with the above property are obtained depending on the geometric conditions on the support of \(\mathfrak{m}\), as well as on the conditions on the field \(k\). These results were applied to the problem of the periodicity of a continuous fraction decomposition constructed in the field of formal power series \(k((1{\text{/}}x))\), for the special elements of the field of functions \(k(\tilde {\mathcal{C}})\) of the hyperelliptic curve \(\tilde {\mathcal{C}}:{{y}^{2}} = f(x)\).

A note on badly approximable systems of linear forms over a field of formal series

We prove that the set of badly approximable systems of M linear forms in N variables over the field of formal power series is hyperplane absolute winning. A consequence of this winning property is that the set has full Hausdorff dimension.

On Mahler's Classification of Formal Power Series Over a Finite Field

Abstract Let K be a finite field, K(x) be the field of rational functions in x over K and K be the field of formal power series over K. We show that under certain conditions integral combinations with algebraic formal power series coefficients of a U 1-number in K are Um -numbers in K , where m is the degree of the algebraic extension of K(x), determined by these algebraic formal power series coefficients.

On the classification problem for polynomials with a periodic continued fraction expansion of in hyperelliptic fields

The classical problem of the periodicity of continued fractions for elements of hyperelliptic fields has a long and deep history. This problem has up to now been far from completely solved. A surprising result was obtained in [1] for quadratic extensions defined by cubic polynomials with coefficients in the field of rational numbers: except for trivial cases there are only three (up to equivalence) cubic polynomials over whose square root has a periodic continued fraction expansion in the field of formal power series. In view of the results in [1], we completely solve the classification problem for polynomials with periodic continued fraction expansion of in elliptic fields with the field of rational numbers as the field of constants.

ON TRANSCENDENTAL CONTINUED FRACTIONS IN FIELDS OF FORMAL POWER SERIES OVER FINITE FIELDS

AbstractIn the field of formal power series over a finite field, we prove a result which enables us to construct explicit examples of $U_{m}$ -numbers by using continued fraction expansions of algebraic formal power series of degree $m>1$ .

A Transcendental Unbounded Continued Fraction Expansions over A Finite Field

Let Fq be a finite field and Fq((X−1 )) the field of formal power series with coefficients in Fq. The purpose of this paper is to exhibit a family of transcendental continued fractions of formal power series over a finite field through some specific irregularities of its partial quotients

Equidistribution Modulo 1

The generalisation of questions of the classic arithmetic has long been of interest. One line of questioning, introduced by Car in 1995, inspired by the equidistribution of the sequence n α n ∈ N where 0 < α < 1 , is the study of the sequence K 1 / l , where K is a polynomial having an l-th root in the field of formal power series. In this paper, we consider the sequence L ′ 1 / l , where L ′ is a polynomial having an l-th root in the field of formal power series and satisfying L ′ ≡ B mod C . Our main result is to prove the uniform distribution in the Laurent series case. Particularly, we prove the case for irreducible polynomials.

Spectrum of a linear differential equation over a field of formal power series

In this paper we associate to a linear differential equation with coefficients in the field of Laurent formal power series a new geometric object, a spectrum in the sense of Berkovich. We compute this spectrum and show that it contains interesting information about the equation.

Finding both, the continued fraction and the Laurent series expansion of golden ratio analogs in the field of formal power series

The focus of this paper is on formal power series analogs of the golden ratio. We are interested in both their continued fractions expansions as well as their Laurent series expansions. Knowing both of them is for instance important for the study of Kronecker type sequences in the theory of uniform distribution.Our approach studies the Hankel matrices that are determined using the coefficients of the Laurent series expansions. We discover that both matrices in the LU decomposition of these Hankel matrices can be described by a simple recursive algorithm based on the continued fraction coefficients of the golden ratio analogs.The upper triangular factor U possesses several nice properties. First, its entries in the special case where the golden ratio analog is [0;X‾] can be given by using the Catalan's triangular numbers. Nicely, this relation together with our findings on the Hankel matrices are used to derive combinatorial identities involving Catalan's triangular numbers. As a side product we obtain a description of the distribution of the Catalan numbers modulo a prime number.Second, the upper triangular factor U is the columnwise composition of the Zeckendorf-type representations of the powers of X. This Zeckendorf representation for polynomials is introduced in the style of the Zeckendorf representation of the natural numbers and it is based on the Fibonacci polynomials instead of Fibonacci numbers.Third, in the case of positive characteristic, for each purely periodic golden ratio analog we detect a self-similar pattern in the upper triangular factor U of its Hankel matrix. We derive this pattern from a binomial type formula for two non-commutative matrices.Finally, we exemplary show, how this self-similar pattern in the upper triangular factor U can be used to describe the Laurent series expansion of the golden ratio analog [0;X‾].

On the Finiteness of the Number of Expansions into a Continued Fraction of $$\sqrt f $$ for Cubic Polynomials over Algebraic Number Fields

We obtain a complete description of cubic polynomials f over algebraic number fields $$\mathbb{K}$$ of degree $$3$$ over $$\mathbb{Q}$$ for which the 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 $$f \in K[x]$$ with a periodic expansion of $$\sqrt f $$ for extensions of $$\mathbb{Q}$$ of degree at most 6. Additionally, we give a complete description of such polynomials f over an arbitrary field corresponding to elliptic fields with a torsion point of order $$N \geqslant 30$$ .

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.

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.

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].