Monomial discrete valuations in k[[X]]

This paper deals with valuations of fields of formal meromorphic functions and their residue fields. We explicitly describe the residue fields of the monomial valuations. We also classify all the discrete rank one valuations of fields of power series in two and three variables, according to their residue fields. We prove that all our cases are possible and give explicit constructions.

Let $K$ be a field equipped with a discrete valuation $v$. In a pioneering work, S. MacLane determined all valuations on $K(x)$ extending $v$. His work was recently reviewed and generalized by M. Vaqui\'e, by using the graded algebra of a valuation. We extend Vaqui\'e's approach by studying residual ideals of the graded algebra of a valuation as an abstract counterpart of certain residual polynomials which play a key role in the computational applications of the theory. As a ...

Let $(K,v)$ be a valued field, and $\mu$ an inductive valuation on $K[x]$ extending $v$. Let $G_\mu$ be the graded algebra of $\mu$ over $K[x]$, and $\kappa$ the maximal subfield of the subring of $G_\mu$ formed by the homogeneous elements of degree zero. In this paper, we find an algorithm to compute the field $\kappa$ and the residual polynomial operator $R_\mu : K[x]\to\kappa[y]$, where $y$ is another indeterminate, without any need to perform computations in the graded ...

Let $v$ be a rank-one discrete valuation of the field $k((\X))$. We know, after \cite{Bri2}, that if $n=2$ then the dimension of $v$ is 1 and if $v$ is the usual order function over $k((\X))$ its dimension is $n-1$. In this paper we prove that, in the general case, the dimension of a rank-one discrete valuation can be any number between 1 and $n-1$.

In this paper we study the rank one discrete valuations of $k((X_1,... ,X_n))$ whose center in $k\lcor\X\rcor$ is the maximal ideal $(\X)$. In sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in section 5...

In this paper we study the rank one discrete valuations of the field $k((X_1,..., X_n))$ whose center in $k\lcor\X\rcor$ is the maximal ideal. In sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in sectio...

Suppose that $(K,v_0)$ is a valued field, $f(x)\in K[x]$ is a monic and irreducible polynomial and $(L,v)$ is an extension of valued fields, where $L=K[x]/(f(x))$. Let $A$ be a local domain with quotient field $K$ dominated by the valuation ring of $v_0$ and such that $f(x)$ is in $A[x]$. The study of these extensions is a classical subject. This paper is devoted to the problem of describing the structure of the associated graded ring ${\rm gr}_v A[x]/(f(x))$ of $A[x]/(f(x)...

Let $K$ be a field, $\mathcal {O}_v$ a valuation ring of $K$ associated to a valuation $v$: $K\rightarrow\Gamma\cup\{\infty\}$, and ${\bf m}_v$ the unique maximal ideal of $\mathcal {O}_v$. Consider an ideal $\mathcal {I}$ of the free $K$-algebra $K\langle X\rangle =K\langle X_1,...,X_n\rangle$ on $X_1,...,X_n$. If ${\cal I}$ is generated by a subset $\mathcal {G}\subset{\cal O}_v\langle X\rangle$ which is a monic Gr\"obner basis of ${\cal I}$ in $K\langle X\rangle$, where $\...

This article is a natural construction of our previous works. In this article, we employ similar ideas due to MacLane to provide an estimate of IC(K(X)|K,v) when (K(X)|K,v) is a valuation algebraic extension. Our central result is an analogue of the estimate obtained in the valuation transcendental case. We further provide a natural construction of a complete sequence of key polynomials for v over K in the setting of valuation algebraic extensions.

Let $(K,v)$ be a valued field. We review some results of MacLane and Vaqui\'e on extensions of $v$ to valuations on the polynomial ring $K[x]$. We introduce certain MacLane-Vaqui\'e chains of residually transcendental valuations, and we prove that every valuation $\mu$ on $K[x]$ is a limit of a finite or countably infinite MacLane-Vaqui\'e chain. This chain underlying $\mu$ is essentially unique and contains arithmetic data yielding an explicit description of the graded algeb...