Related papers: Optimal Parametrizations and Valuations
This article is a survey of conjectures and results on reductive algebraic groups having good reduction at a suitable set of discrete valuations of the base field. Until recently, this subject has received relatively little attention, but…
A new criterion is given for a semigroup to be the semigroup of a valuation dominating an equicharacteristic local domain. The criterion is used to construct examples of well ordered subsemigroups of the positive rational numbers which are…
We consider ideals $I$ in a Stanley-Reisner ring $k[\Delta]$ over the simplical complex $\Delta$, such that the tight closure of $I$, $I^*$, is equal to $\mathfrak{m}$, the standard graded maximal ideal of $k[\Delta]$. We determine the…
We develop the basic theory of geometrically closed rings as a generalisation of algebraically closed fields, on the grounds of notions coming from positive model theory and affine algebraic geometry. For this purpose we consider several…
This paper extends the article of the Bruns and Conca on SAGBI bases and their computation (J. Symb. Comput. 120 (2024)) in two directions. (i) We describe the extension of the Singular library sagbiNormaliz.sing to the computation of…
It is shown that every commutative local ring of bounded module type is an almost maximal valuation ring.
We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial…
Given a noetherian local domain $R$ and a valuation $\nu$ of its field of fractions which is non negative on $R$, we derive some very general bounds on the growth of the number of distinct valuation ideals of $R$ corresponding to values…
We give a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. We also discuss various generalizations of this problem and some…
A \textit{symmetric ideal} $I \subseteq R = K[x_1,x_2,...]$ is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gr\"obner bases for symmetric ideals in the infinite…
For a commutative ring R we investigate the property that the sets of minimal primes of finitely generated ideals of R is always finite. We prove this property passes to polynomial ring extensions (in an arbitrary number of variables) over…
Suppose that $k$ is an arbitrary field. Consider the field $k((x_1,...,x_n))$, which is the quotient field of the ring $k[[x_1,...,x_n]]$ of formal power series in the variables $x_1,...,x_n$, with coefficients in $k$. Suppose that $\sigma$…
We investigate whether the group algebra of a finite group over a localisation of the integers is semiperfect. The main result is a necessary and sufficient arithmetic criterion in the ordinary case. In the modular case, we propose a…
The aim of this work is to study duality of fractional ideals with respect to a fixed ideal and to investigate the relationship between value sets of pairs of dual ideals in admissible rings, a class of rings that contains the local rings…
We interpret a valuation $v$ on a ring $R$ as a map $v: R \to M$ into a so called bipotent semiring $M$ (the usual max-plus setting), and then define a \textbf{supervaluation} $\phi$ as a suitable map into a supertropical semiring $U$ with…
Consider the problem of minimizing a lower semi-continuous semi-algebraic function $f \colon \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ on an unbounded closed semi-algebraic set $S \subset \mathbb{R}^n.$ Employing adequate tools of…
The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gr\"obner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent…
The arithmetic rank of an ideal in a polynomial ring over an algebraically closed field is the smallest number of equations needed to define its vanishing locus set-theoretically. We determine the arithmetic rank of the generic $m$-residual…
We study the ridge method for min-max problems, and investigate its convergence without any convexity, differentiability or qualification assumption. The central issue is to determine whether the ''parametric optimality formula'' provides a…
It is shown that a local ring R of bounded module type is an almost maximal valuation ring if there exists a non-maximal prime ideal J such that R/J is a maximal valuation domain.