Related papers: Key Polynomials in dimension 2

We make two observations regarding the invertibility of Keller maps. i.e., polynomial maps for which the determinant of their Jacobian matrix is identically equal to 1. In our first result, we show that if P is a n-dimensional Keller map,…

In this paper we complete the proof of the "equivalence" of non-discrete R-buildings of types A~_2 and C~_2, with, respectively, projective planes and generalized quadrangles with non-discrete valuation, begun in previous paper of the…

Rings of integer-valued polynomials are known to be atomic, non-factorial rings furnishing examples for both irreducible elements for which all powers factor uniquely (\emph{absolutely irreducibles}) and irreducible elements where some…

We provide an irreducibility test in the ring K[[x]][y] whose complexity is quasi-linear with respect to the discriminant valuation, assuming the input polynomial F square-free and K a perfect field of characteristic zero or greater than…

In this paper we study the truncation $\nu_q$ of a valuation $\nu$ on a polynomial $q$. It is known that when $q$ is a key polynomial, then $\nu_q$ is a valuation. It is also known that the converse does not hold. We show that when $q$ is a…

We describe the set V of all real valued valuations v on the ring C[[x,y]] normalized by min{v(x),v(y)}=1. It has a natural structure of an R-tree, induced by the order relation v is less than v' iff v(f) is less than v'(f) for all f. It…

We study a graded vector space of polynomials associated to a square matrix, defined by a finite difference condition along the rows. We show this space coincides with one defined by directional derivatives, and prove it is…

A skew polynomial ring $R=K[x;\sigma,\delta]$ is a ring of polynomials with non-commutative multiplication. This creates a difference between left and right divisibility, and thus a concept of left and right evaluations and roots. A…

Let R be an affine k-domain over the field k. The paper's main result is that, if R admits a non-trivial embedding in a polynomial ring K[s] for some field K containing k, then R can be embedded in a polynomial ring F[t] which extends R…

The Weyl algebra over a field $k$ of characteristic $0$ is a simple ring of Gelfand-Kirillov dimension 2, which has a grading by the group of integers. We classify all $\mathbb{Z}$-graded simple rings of GK-dimension 2 and show that they…

Let $\nu$ be a rank one valuation on $K[x]$ and $\Psi_n$ the set of key polynomials for $\nu$ of degree $n\in\N$. We discuss the concepts of being $\Psi_n$-stable and $(\Psi_n,Q)$-fixed. We discuss when these two concepts coincide. We use…

Within this research, two combinatorial bijections using Young diagrams were studied. The first is a special case of a bijective correspondence between two classes of combinatorial objects. Its proof, based on Young diagrams, establishes…

Let $V$ be a valuation domain with quotient field $K$. We show how to describe all extensions of $V$ to $K(X)$ when the $V$-adic completion $\widehat{K}$ is algebraically closed, generalizing a similar result obtained by Ostrowski in the…

Let $D$ be a 2-dimensional regular local ring and let $Q(D)$ denote the quadratic tree of 2-dimensional regular local overrings of $D$. We examine the Noetherian rings that are intersections of rings in $Q(D)$. To do so, we describe the…

Let $(K,\nu)$ be an arbitrary valued field with valuation ring $R_{\nu}$ and $L=K(\alpha)$, where $\alpha$ is a root of a monic irreducible polynomial $f\in R_{\nu}[x]$. In this paper, we characterize the integral closedness of…

We characterize the polynomial closure of a pseudo-convergent sequence in a valuation domain $V$ of arbitrary rank, and then we use this result to show that the polynomial closure is never topological when $V$ has rank at least $2$.

We introduce a new concept of rank - relative rank associated to a filtered collection of polynomials. When the filtration is trivial our relative rank coincides with Schmidt rank (also called strength). We also introduce the notion of…

We define a $q$-deformation of the classical ring of integer-valued polynomials which we call the ring of quantum integer-valued polynomials. We show that this ring has a remarkable combinatorial structure and enjoys many positivity…

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 give a first-order definition of key polynomials, we show the links with previous definitions, that it is relevant to study key degrees, and to use a kind of valuations that we call partially multiplicative. We also prove or reprove…