Field Of Power Series Research Articles

CONTINUED FRACTION AND DIOPHANTINE EQUATION

Abstract. Our paper is devoted to the study of certain diophantineequations on the ring of polynomials over a finite field, which are inti-mately related to algebraic formal power series which have partial quo-tients of unbounded degree in their continued fraction expansion. Inparticular it is shown that there are Pisot formal power series with de-gree greater than 2, having infinitely many large partial quotients in theirsimple continued fraction expansions. This generalizes an earlier resultof Baum and Sweet for algebraic formal power series. 1. IntroductionLet F q be a finite field of characteristic p with q elements. We considerF q [X], F q (X) and F q ((X −1 )) as analogues of Z, Q and R respectively. Ifw =P +∞n=n 0 a n X −n is an element of F q ((X −1 )) with a n 0 6= 0, we introducethe absolute value |w| = q −n 0 and |0| = 0. Diophantine approximation inthe function field case was initiated by K. Mahler [10]. In the case of realnumbers, it is well known that Liouville’s theorem was the beginning of along path, with the works of Thue, Siegel, Dyson and others, leading of thecelebrated Roth’s theorem which was established in 1955. This improvementcan be transposed in fields of power series if the base field has characteristiczero, as shown by Uchiyama in 1960. But this is not the case in positivecharacteristicand consequentlythe studyofrationalapproximationtoalgebraicelements becomes somewhat more complex.Many examples can be studied. A special case is the one where the algebraicelement w satisfies an equation of the form w =

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

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

Symbolic computation of Drazin inverses by specializations

In this paper, we show how to reduce the computation of Drazin inverses over certain computable fields to the computation of Drazin inverses of matrices with rational functions as entries. As a consequence we derive a symbolic algorithm to compute the Drazin inverse of matrices whose entries are elements of a finite transcendental field extension of a computable field. The algorithm is applied to matrices over the field of meromorphic functions, in several complex variables, on a connected domain and to matrices over the field of Laurent formal power series.

Pisot-K Elements in the Field of Formal Power Series over Finite Field

In this paper, we will give a criteria of irreducible polynomials over i†q[X] where i†q is a finite field. We will present an estimation for the number of the Pisot-k power formal series, precisely we will give their degrees and their logarithmic heights

Symmetrically complete ordered sets abelian groups and fields

We characterize linearly ordered sets, abelian groups and fields that are symmetrically complete, meaning that the intersection over any chain of closed bounded intervals is nonempty. Such ordered abelian groups and fields are important because generalizations of Banach’s Fixed Point Theorem hold in them. We prove that symmetrically complete ordered abelian groups and fields are divisible Hahn products and real closed power series fields, respectively. This gives us a direct route to the construction of symmetrically complete ordered abelian groups and fields, modulo an analogous construction at the level of ordered sets; in particular, this gives an alternative approach to the construction of symmetrically complete fields in [12].

Multiplicatively badly approximable matrices in fields of power series

Multiplicatively badly approximable matrices in fields of power series

The -cyclic McKay correspondence via motivic integration

AbstractWe study the McKay correspondence for representations of the cyclic group of order $p$ in characteristic $p$. The main tool is the motivic integration generalized to quotient stacks associated to representations. Our version of the change of variables formula leads to an explicit computation of the stringy invariant of the quotient variety. A consequence is that a crepant resolution of the quotient variety (if any) has topological Euler characteristic $p$ as in the tame case. Also, we link a crepant resolution with a count of Artin–Schreier extensions of the power series field with respect to weights determined by ramification jumps and the representation.

A note on the real densities of homogeneous systems in function fields

Let Fq((1/t)) denote the field of formal power series in 1/t over the finite field Fq of q elements. In this paper, we prove the existence of the real densities of certain homogeneous systems in Fq((1/t)). In addition, we show that the real density of a system is equal to the corresponding singular integral.

The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field

Abstract.Dufresnoy and Pisot characterized the smallest Pisot number of degree n ≥ 3 by giving explicitly its minimal polynomial. In this paper, we translate Dufresnoy and Pisot’s result to the Laurent series case. The aim of this paper is to prove that the minimal polynomial of the smallest Pisot element (SPE) of degree n in the field of formal power series over a finite field is given by P(Y) = Yn–XYn-1–αn where α is the least element of the finite field 픽q\{0} (as a finite total ordered set). We prove that the sequence of SPEs of degree n is decreasing and converges to αX: Finally, we show how to obtain explicit continued fraction expansion of the smallest Pisot element over a finite field.

A note on Schanuel’s conjectures for exponential logarithmic power series fields

In Ax (Ann. Math. 93(2):252–268, 1971), J. Ax proved a transcendency theorem for certain differential fields of characteristic zero : the differential counterpart of the still open Schanuel conjecture about the exponential function over \({\mathbb{C}}\) (Lang, Introduction to transcendental numbers, 1966). In this article, we derive from Ax’s theorem transcendency results in the context of differential valued exponential fields. In particular, we obtain results for exponential Hardy fields, Logarithmic-Exponential power series fields, and Exponential-Logarithmic power series fields.

Klein approximation and Hilbertian fields

The quotient field of a generalized Krull domain of dimension exceeding one is Hilbertian by a theorem of Weissauer. Building on work of R. Klein we generalize this criterion for Hilbertianity to a wider class of domains. This allows us to extend recent results on Hilbertianity of fields of power series and obtain new Hilbertian fields.

$q$-Conjugacy classes in loop groups

Let k((t)) be the field of formal Laurent power series over some field k of characteristic zero, q ∈ k which is not a root of unity, and σq be the automorphism of k((t)) defined by σq : α(t) 7→ α(qt). For an algebraic group G over k write G(k((t))) for the k((t))-points of G. Then σq also acts on G(k((t))) by change of variable, namely σq(g(t)) = g(qt). One can define a q-conjugate action of G(k((t))) on itself by

Hilbertianity of fields of power series

AbstractLetRbe a domain contained in a rank-1 valuation ring of its quotient field. LetR⟦X⟧ be the ring of formal power series overR, and letFbe the quotient field ofR⟦X⟧. We prove thatFis Hilbertian. This resolves and generalizes an open problem of Jarden, and allows to generalize previous Galois-theoretic results over fields of power series.

Schanuel property for additive power series

We prove a version of Schanuel’s Conjecture for a field of Laurent power series in positive characteristic replacing ℂ and a non-algebraic additive power series replacing the exponential map.

A note on multiple exponential sums in function fields

Let A = F q [ t ] denote the ring of polynomials over the finite field F q . We denote by e a certain non-trivial character of the field of formal power series in terms of 1 / t over F q . For a monic g ∈ A and a polynomial G ( x 1 , … , x d ) ∈ A [ x 1 , … , x d ] , we define S ( G ; g ) = ∑ x e ( G ( x ) / g ) where x runs over ( A / ( g ) ) d . In this paper, we prove that if G ( x 1 , … , x d ) is a homogeneous polynomial of degree k and if the g.c.d. of g and the coefficients of G is 1, then S ( G ; g ) ≪ 〈 g 〉 d − 1 2 k + ϵ , where 〈 g 〉 = q deg g and ϵ is a small positive constant.

Maps on Ultrametric Spaces, Hensel's Lemma, and Differential Equations Over Valued Fields

We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multidimensional Hensel's Lemma follow from our result. We give an easy proof that the latter holds in every henselian field. We also prove a basic infinite-dimensional Implicit Function Theorem. Further, we apply the criterion to deduce various versions of Hensel's Lemma for polynomials in several additive operators, and to give a criterion for the existence of integration and solutions of certain differential equations on spherically complete valued differential fields, for both valued D-fields in the sense of Scanlon, and differentially valued fields in the sense of Rosenlicht. We modify the approach so that it also covers logarithmic-exponential power series fields. Finally, we give a criterion for a sum of spherically complete subgroups of a valued abelian group to be spherically complete. This in turn can be used to determine elementary properties of power series fields in positive characteristic.

Cyclic 2-structures and spaces of orderings of power series fields in two variables

We consider the space of orderings of the field R ( ( x , y ) ) and the space of orderings of the field R ( ( x ) ) ( y ) , where R is a real closed field. We examine the structure of these objects and their relationship to each other. We define a cyclic 2-structure to be a pair ( S , Φ ) where S is a cyclically ordered set and Φ is an equivalence relation on S such that each equivalence class has exactly two elements. We show that each of these spaces of orderings is described by a cyclic 2-structure, in a natural way. We also show that if the real closed field R is archimedean then the space of R -places of these fields is describable in terms of the cyclic 2-structure.

ON PERIODIC CONTINUED FRACTIONS OVER $\mathbb{F}_q((X^{−1}))$

Let $\mathbb{F}_q$ be a field with $q$ elements of characteristic $p$ and $\mathbb{F}_q((X^{−1}))$ be the field of formal power series over $\mathbb{F}_q$. Let $f$ be a quadratic formal power series of continued fraction expansion $[b_0; b_1, \ldots, b_s, \overline{a_1, \ldots, a_t}]$, we denote by $t = \operatorname{Per}(f)$ the period length of the partial quotients of $f$. The aim of this paper is to study the continued fraction expansion of $Af$ where $A$ is a polynomial $\in \mathbb{F}_q[X]$. In particular we study the asymptotic behavior of the functions \[ S(N,n) = \sup_{\operatorname{deg} A = N} \sup_{f \in \Lambda_{n}} \operatorname{Per}(Af) \quad \textrm{and} \quad R(N) = \sup_{n \geq 1} \frac{S(N,n)}{n}, \] where $\Lambda_{n}$ is the set of quadratic formal power series of period $n$ in $\mathbb{F}_q((X^{−1}))$.

On the Quadratic Continued Fractions Over 𝔽 Q (X)

Let 𝔽 q ((X −1)) be the field of formal power series in X −1 over 𝔽 q , the field with q elements. Let f ∈ 𝔽 q ((X −1)) satisfy the irreducible polynomial Af 2 + Bf + C = 0, with Δ =B 2 − 4AC. Let Per(f) be the length of the period of the continued fraction expansion of f. In this article, we show that We also prove that if Q is a monic polynomial with even degree, then the length of the period of the continued fraction expansion of any square root of Q is less than

A GALOIS THEORY FOR THE FIELD EXTENSION K((X))/K

AbstractLet K be a field of characteristic 0, which is algebraically closed to radicals. Let F = K((X)) be the valued field of Laurent power series and let G = Aut(F/K). We prove that if L is a subfield of F, K ≠ L, such that L/K is a sub-extension of F/K and F/L is a Galois algebraic extension (L/K is Galois coalgebraic in F/K), then L is closed in F, F/L is a finite extension and Gal(F/L) is a finite cyclic group of G. We also prove that there is a one-to-one and onto correspondence between the set of all finite subgroups of G and the set of all Galois coalgebraic sub-extensions of F/K. Some other auxiliary results which are useful by their own are given.