The close relation between border and Pommaret marked bases

Given a finite order ideal $${\mathcal {O}}$$ O in the polynomial ring $$K[x_1,\ldots , x_n]$$ K [ x 1 , … , x n ] over a field K, let $$\partial {\mathcal {O}}$$ ∂ O be the border of $${\mathcal {O}}$$ O and $${\mathcal {P}}_{\mathcal {O}}$$ P O the Pommaret basis of the ideal generated by the terms outside $${\mathcal {O}}$$ O . In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $$\partial {\mathcal {O}}$$ ∂ O -marked sets (resp. bases) and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked sets (resp. bases). We prove that a $$\partial {\mathcal {O}}$$ ∂ O -marked set B is a marked basis if and only if the $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked set P contained in B is a marked basis and generates the same ideal as B. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $$\partial {\mathcal {O}}$$ ∂ O -marked bases and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gröbner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in affine spaces of lower dimension. Furthermore, we observe that Pommaret marked schemes give an open covering of Hilbert schemes parameterizing 0-dimensional schemes without any group actions. Several examples are given throughout the paper.

Computing subschemes of the border basis scheme

A good way of parameterizing zero-dimensional schemes in an affine space [Formula: see text] has been developed in the last 20 years using border basis schemes. Given a multiplicity [Formula: see text], they provide an open covering of the Hilbert scheme [Formula: see text] and can be described by easily computable quadratic equations. A natural question arises on how to determine loci which are contained in border basis schemes and whose rational points represent zero-dimensional [Formula: see text]-algebras sharing a given property. The main focus of this paper is on giving effective answers to this general problem. The properties considered here are the locally Gorenstein, strict Gorenstein, strict complete intersection, Cayley–Bacharach, and strict Cayley–Bacharach properties. The key characteristic of our approach is that we describe these loci by exhibiting explicit algorithms to compute their defining ideals. All results are illustrated by nontrivial, concrete examples.

Computation of the unipotent Albanese map on elliptic and hyperelliptic curves

We study the unipotent Albanese map appearing in the non-abelian Chabauty method of Minhyong Kim. In particular we explore the explicit computation of the p-adic de Rham period map $$j^{dr}_n$$ j n dr on elliptic and hyperelliptic curves over number fields via their universal unipotent connections $${\mathscr {U}}$$ U . Several algorithms forming part of the computation of finite level versions $$j^{dr}_n$$ j n dr of the unipotent Albanese maps are presented. The computation of the logarithmic extension of $${\mathscr {U}}$$ U in general requires a description in terms of an open covering, and can be regarded as a simple example of computational descent theory. We also demonstrate a constructive version of a lemma of Hadian used in the computation of the Hodge filtration on $${\mathscr {U}}$$ U over affine elliptic and odd hyperelliptic curves. We use these algorithms to present some new examples describing the co-ordinates of some of these period maps. This description will be given in terms iterated p-adic Coleman integrals. We also consider the computation of the co-ordinates if we replace the rational basepoint with a tangential basepoint, and present some new examples here as well.

STABILITY ANALYSIS OF PARAMETRIC GENERALIZED EQUATIONS AND APPLICATIONS

The paper studies the stability of parametric generalized equations (PGE) defined by adding two set-valued mappings. Under some moderate conditions, the open covering/metric regularity of the solution mapping of PGE are discussed. The results are used to analyze the stability of stochastic generalized equations and parametric mathematical programs with equilibrium constraints.

Covering analytic sets by families of closed set

We prove that for every familyIof closed subsets of a Polish space eachset can be covered by countably many members ofIor else contains a nonemptyset which cannot be covered by countably many members ofI. We prove an analogous result forκ-Souslin sets and show that ifA#exists for anyA⊂ωω, then the above result is true forsets. A theorem of Martin is included stating that this result is also true for weakly homogeneously Souslin sets. As an application of our results we derive from them a general form of Hurewicz's theorem due to Kechris. Louveau, and Woodin and a theorem of Feng on the open covering axiom. Also some well-known theorems on finding “big” closed sets insideof“big”andare consequences of our results.

On Σ-Finite Families

Let be a family of subsets of a topological space X. We do not require to be a covering of X, nor do we assume that the members of are necessarily open. In this paper we shall assume that is of a special sort, which we call Σ-Finite. We show that a Σ-Finite family is both locally finite and star-finite, and in particular that an open covering of X is Σ-Finite if and only if it is star-finite. We then prove that every Σ-Finite family is ᓂ-discrete, so that in particular, every star-finite open covering of X is (ᓂ-discrete. There seems to be some applications of this fact.

A Note on Certain Spaces with Bases (mod K)

In this note all spaces are assumed to be regular T1 spaces and all undefined terms and notations may be found in [8], In particular let cl(A) denote the closure of the set A and let Z+ denote the set of natural numbers.Definition 1. Let X be a topological space and a covering of X by compact sets. An open covering of X is said to be a basis (mod K) if whenever and an open set V contains Kx, then there exists such that . In such a case X is written as the ordered triple .

On irreducible spaces

A constructive proof of a theorem of Worrell and Wicke, which states that every open covering of a θ-refinable space has a σ-distributively point-finite open refinement that covers the space minimally, is presented. Spaces for which every open covering has an open refinement which covers the space minimally are characterized by the use of discrete closed collections and some open questions relating to spaces of this type are included.