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On Newton's polygons, Gröbner bases and series expansions of perturbed polynomial programs

Konstantin Avrachenkov, Vladimir Ejov, Jerzy A. Filar (2006)

Banach Center Publications

In this note we consider a perturbed mathematical programming problem where both the objective and the constraint functions are polynomial in all underlying decision variables and in the perturbation parameter ε. Recently, the theory of Gröbner bases was used to show that solutions of the system of first order optimality conditions can be represented as Puiseux series in ε in a neighbourhood of ε = 0. In this paper we show that the determination of the branching order and the order of the pole (if...

On prime modules over pullback rings

Shahabaddin Ebrahimi Atani (2004)

Czechoslovak Mathematical Journal

First, we give a complete description of the indecomposable prime modules over a Dedekind domain. Second, if R is the pullback, in the sense of [9], of two local Dedekind domains then we classify indecomposable prime R -modules and establish a connection between the prime modules and the pure-injective modules (also representable modules) over such rings.

On prime submodules and primary decomposition

Yücel Tiraş, Harmanci, Abdullah (2000)

Czechoslovak Mathematical Journal

We characterize prime submodules of R × R for a principal ideal domain R and investigate the primary decomposition of any submodule into primary submodules of R × R .

On prolongations of rank one discrete valuations

Lhoussain El Fadil (2019)

Commentationes Mathematicae Universitatis Carolinae

Let ( K , ν ) be a valued field, where ν is a rank one discrete valuation. Let R be its ring of valuation, 𝔪 its maximal ideal, and L an extension of K , defined by a monic irreducible polynomial F ( X ) ∈ R [ X ] . Assume that F ¯ ( X ) factors as a product of r distinct powers of monic irreducible polynomials. In this paper a condition which guarantees the existence of exactly r distinct valuations of K extending ν is given, in such a way that it generalizes the results given in the paper “Prolongations of valuations to finite...

On proximity relations for valuations dominating a two-dimensional regular local ring.

José J. Aparicio, Angel Granja, Tomás Sánchez-Giralda (1999)

Revista Matemática Iberoamericana

The purpose of this paper is to define a new numerical invariant of valuations centered in a regular two-dimensional regular local ring. For this, we define a sequence of non-negative rational numbers δν = {δν(j)}j ≥ 0 which is determined by the proximity relations of the successive quadratic transformations at the points determined by a valuation ν. This sequence is characterized by seven combinatorial properties, so that any sequence of non-negative rational numbers having the above properties...

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