Related papers: Valuations in algebraic field extensions

We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials $f(x,y)$ over an arbitrary field $\mathbb{K}$. Our results rely on information on the…

We study the finite field extension estimates for Hamming varieties $H_j, j\in \mathbb F_q^*,$ defined by $H_j=\{x\in \mathbb F_q^d: \prod_{k=1}^d x_k=j\},$ where $\mathbb F_q^d$ denotes the $d$-dimensional vector space over a finite field…

This paper deals with properties of the algebraic variety defined as the set of zeros of a "deficient" sequence of multivariate polynomials. We consider two types of varieties: ideal-theoretic complete intersections and absolutely…

Smoktunowicz, Lenagan, and the second-named author recently gave an example of a nil algebra of Gelfand-Kirillov dimension at most three. Their construction requires a countable base field, however. We show that for any field $k$ and any…

Let $\mathcal{R}$ be a finite valuation ring of order $q^r$. In this paper, we prove that for any quadratic polynomial $f(x,y,z) \in \mathcal{R}[x,y,z]$ that is of the form $axy+R(x)+S(y)+T(z)$ for some one-variable polynomials $R, S , T$,…

We provide upper bounds for the cardinality of the value set of a polynomial map in several variables over a finite field. These bounds generalize earlier bounds for univariate polynomials.

Let K be a field with a valuation $\nu$ and let L = K(x) be a transcendental extension of K, then any valuation $\mu$ of L which extends $\nu$ is determined by its restriction to the polynomial ring K[x]. We know how to associate to this…

For a field $K$, and a root $\alpha$ of an irreducible polynomial over $K$ (in some algebraic closure) the number of roots of $f(x)$ lying in $K(\alpha)$ is studied here. Given such an $f(x)$ of degree $n$ for which $r$ of the roots are i n…

We introduce the notion of {\it approximation type} for the partial, and in certain cases the total description of extensions of a given valuation from a field $K$ to the rational function field $K(x)$. To every extension, a unique…

In different areas of discrete mathematics, a certain type of polynomials, having coefficients in a field K of finite characteristic, has been considered. The form and the degree of these polynomials, here called projective, are simply…

This paper analyzes theorems about algebraic field extensions using the techniques of reverse mathematics. In section 2, we show that $\mathsf{WKL}_0$ is equivalent to the ability to extend $F$-automorphisms of field extensions to…

Let $K$ be an algebraically closed field of characteristic zero and ${P_n=K[x_1,\ldots,x_n]}$ the polynomial ring. Any $K$-derivation $D$ on $P_n$ is of the form ${ D=\sum_{i=1}^n f_i(x_1,\ldots,x_n)\frac{\partial}{\partial x_i} },$ where…

Assume that $X= {x_1,...,x_g}$ is a finite alphabet and $K$ is a field. We study monomial algebras $A= K <X> /(W)$, where $W$ is an antichain of Lyndon words in $X$ of arbitrary cardinality. We find a Poincar\'{e}-Birkhoff-Witt type basis…

Given a global field K and a polynomial f defined over K of degree at least two, Morton and Silverman conjectured in 1994 that the number of K-rational preperiodic points of f is bounded in terms of only the degree of K and the degree of f.…

A classical tool in the study of real closed fields are the fields $K((G))$ of generalized power series (i.e., formal sums with well-ordered support) with coefficients in a field $K$ of characteristic 0 and exponents in an ordered abelian…

In this essay we explore the notion of essential dimension using the theory of valuations of fields. Given a field extension K/k and a valuation on K that is trivial on k, we prove that the rank of the valuation cannot exceed the…

The question about maximal size of independent system of word equations is one of the most striking problems in combinatorics on words. Recently, Aleksi Saarela has introduced a new approach to the problem that is based on linear-algebraic…

In Dokchitser (2007) it is shown that given an elliptic curve $E$ defined over a number field $K$ then there are infinitely many degree 3 extensions $L/K$ for which the rank of $E(L)$ is larger than $E(K)$. In the present paper we show that…

Let $\iota:(K,\nu)\hookrightarrow(K(x),\mu)$ be a simple purely transcendental extension of valued fields. In order to study such an extension, M. Vaqui\'e, generalizing an earlier construction of S. Mac Lane, introduced the notion of Key…

Suppose L is any finite algebraic extension of either the ordinary rational numbers or the p-adic rational numbers. Also let g_1,...,g_k be polynomials in n variables, with coefficients in L, such that the total number of monomial terms…