Approximation types describing extensions of valuations to rational function fields

The Growth of Valuations on Rational Function Fields

Class numbers of some abelian extensions of rational function fields

Let P be a monic irreducible polynomial. In this paper we generalize the determinant formula for h(K Pn) of Bae and Kang and the formula for h−(KPn ) of Jung and Ahn to any subfields K of the cyclotomic function field KPn . By using these formulas, we calculate the class numbers h −(K), h(K+) of all subfields K of KP when q and deg(P ) are small.

A method to obtain the best uniform polynomial approximation for the family of rational function

In this article, by using Chebyshev’s polynomials and Chebyshev’s expansion, we obtain the best uniform polynomial approximation out of P2n to a class of rational functions of the form (ax2+c)-1 on any non symmetric interval [d,e]. Using the obtained approximation, we provide the best uniform polynomial approximation to a class of rational functions of the form (ax2+bx+c)-1 for both cases b2-4a...

Well-ordered Valuations on Characteristic Zero Rational Function Fields of Transcendence Degree Two

Given a valuation on the function field k(x, y), we examine the set of images of nonzero elements of the underlying polynomial ring k[x, y] under this valuation. For an arbitrary field k, a Noetherian power series is a map z : Q → k that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on k(x, y). Although the value groups correspondi...

Symmetry, splitting rational places in extensions of function fields and generalization of the Hermitian function field

Let F/K be an algebraic function field in one variable over a finite field of constants K, i.e., F is a finite algebraic extension of K(x) where x ∈ F is transcendental over K. Let E be a finite separable extension of F . Let N(E) and g(E) denote the number of places of degree one (or rational places), and the genus, respectively, of E. Let [E : F ] denote the degree of this extension. In recen...