scholarly journals Macaulay-like marked bases

2017 ◽  
Vol 16 (05) ◽  
pp. 1750100 ◽  
Author(s):  
Cristina Bertone ◽  
Francesca Cioffi ◽  
Margherita Roggero
Keyword(s):  
Polynomial Ring ◽  
Open Subset ◽  
Hilbert Scheme ◽  
Combinatorial Properties ◽  
Ideal Formula ◽  
Complexity Results ◽  
The Family ◽  
Stable Ideal ◽  
Affine Scheme

We define marked sets and bases over a quasi-stable ideal [Formula: see text] in a polynomial ring on a Noetherian [Formula: see text]-algebra, with [Formula: see text] a field of any characteristic. The involved polynomials may be non-homogeneous, but their degree is bounded from above by the maximum among the degrees of the terms in the Pommaret basis of [Formula: see text] and a given integer [Formula: see text]. Due to the combinatorial properties of quasi-stable ideals, these bases behave well with respect to homogenization, similarly to Macaulay bases. We prove that the family of marked bases over a given quasi-stable ideal has an affine scheme structure, is flat and, for large enough [Formula: see text], is an open subset of a Hilbert scheme. Our main results lead to algorithms that explicitly construct such a family. We compare our method with similar ones and give some complexity results.

scholarly journals Cotangent spaces and separating re-embeddings

2021 ◽  
pp. 2250188
Author(s):  
Martin Kreuzer ◽  
Le Ngoc Long ◽  
Lorenzo Robbiano
Keyword(s):  
Lower Bound ◽  
Polynomial Ring ◽  
Maximal Ideal ◽  
Embedding Dimension ◽  
Affine Algebra ◽  
Ideal Formula ◽  
Cotangent Space ◽  
Gröbner Fan ◽  
Affine Scheme ◽  
The Ideal

Given an affine algebra [Formula: see text], where [Formula: see text] is a polynomial ring over a field [Formula: see text] and [Formula: see text] is an ideal in [Formula: see text], we study re-embeddings of the affine scheme [Formula: see text], i.e. presentations [Formula: see text] such that [Formula: see text] is a polynomial ring in fewer indeterminates. To find such re-embeddings, we use polynomials [Formula: see text] in the ideal [Formula: see text] which are coherently separating in the sense that they are of the form [Formula: see text] with an indeterminate [Formula: see text] which divides neither a term in the support of [Formula: see text] nor in the support of [Formula: see text] for [Formula: see text]. The possible numbers of such sets of polynomials are shown to be governed by the Gröbner fan of [Formula: see text]. The dimension of the cotangent space of [Formula: see text] at a [Formula: see text]-linear maximal ideal is a lower bound for the embedding dimension, and if we find coherently separating polynomials corresponding to this bound, we know that we have determined the embedding dimension of [Formula: see text] and found an optimal re-embedding.

scholarly journals Further steps on the reconstruction of convex polyominoes from orthogonal projections

2021 ◽  
Author(s):  
Paolo Dulio ◽  
Andrea Frosini ◽  
Simone Rinaldi ◽  
Lama Tarsissi ◽  
Laurent Vuillon
Keyword(s):  
Discrete Geometry ◽  
Convex Subset ◽  
Convex Sets ◽  
A Priori ◽  
Orthogonal Projections ◽  
Reconstruction Process ◽  
Combinatorial Properties ◽  
The Family ◽  
Discrete Counterpart ◽  
Interior Part

AbstractA remarkable family of discrete sets which has recently attracted the attention of the discrete geometry community is the family of convex polyominoes, that are the discrete counterpart of Euclidean convex sets, and combine the constraints of convexity and connectedness. In this paper we study the problem of their reconstruction from orthogonal projections, relying on the approach defined by Barcucci et al. (Theor Comput Sci 155(2):321–347, 1996). In particular, during the reconstruction process it may be necessary to expand a convex subset of the interior part of the polyomino, say the polyomino kernel, by adding points at specific positions of its contour, without losing its convexity. To reach this goal we consider convexity in terms of certain combinatorial properties of the boundary word encoding the polyomino. So, we first show some conditions that allow us to extend the kernel maintaining the convexity. Then, we provide examples where the addition of one or two points causes a loss of convexity, which can be restored by adding other points, whose number and positions cannot be determined a priori.

Ideals Modulo a Prime

2020 ◽  
pp. 2150042
Author(s):  
John Abbott ◽  
Anna Maria Bigatti ◽  
Lorenzo Robbiano
Keyword(s):  
Prime Number ◽  
Polynomial Ring ◽  
Distinct Advantage ◽  
Ideal Formula ◽  
Reduction Modulo ◽  
Good For ◽  
Definition Of ◽  
The Given ◽  
Term Ordering

The main focus of this paper is on the problem of relating an ideal [Formula: see text] in the polynomial ring [Formula: see text] to a corresponding ideal in [Formula: see text] where [Formula: see text] is a prime number; in other words, the reduction modulo[Formula: see text] of [Formula: see text]. We first define a new notion of [Formula: see text]-good prime for [Formula: see text] which does depends on the term ordering [Formula: see text], but not on the given generators of [Formula: see text]. We relate our notion of [Formula: see text]-good primes to some other similar notions already in the literature. Then we introduce and describe a new invariant called the universal denominator which frees our definition of reduction modulo [Formula: see text] from the term ordering, thus letting us show that all but finitely many primes are good for [Formula: see text]. One characteristic of our approach is that it enables us to easily detect some bad primes, a distinct advantage when using modular methods.

scholarly journals Fan-Planar Graphs: Combinatorial Properties and Complexity Results

2014 ◽  
pp. 186-197 ◽  
Author(s):  
Carla Binucci ◽  
Emilio Di Giacomo ◽  
Walter Didimo ◽  
Fabrizio Montecchiani ◽  
Maurizio Patrignani ◽  
...  
Keyword(s):  
Planar Graphs ◽  
Combinatorial Properties ◽  
Complexity Results

On the reduction numbers of monomial ideals

2019 ◽  
Vol 19 (10) ◽  
pp. 2050201
Author(s):  
Ibrahim Al-Ayyoub
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Upper Bounds ◽  
Integral Closure ◽  
Monomial Ideals ◽  
Explicit Method ◽  
Generating Set ◽  
Ideal Formula ◽  
Reduction Number ◽  
The Ideal

Let [Formula: see text] be a monomial ideal in a polynomial ring with two indeterminates over a field. Assume [Formula: see text] is contained in the integral closure of some ideal that is generated by two elements from the generating set of [Formula: see text]. We produce sharp upper bounds for each of the reduction number and the Ratliff–Rush reduction number of the ideal [Formula: see text]. Under certain hypotheses, we give the exact values of these reduction numbers, and we provide an explicit method for obtaining these sharp upper bounds.

scholarly journals 3-Colorability of Pseudo-Triangulations

2015 ◽  
Vol 25 (04) ◽  
pp. 283-298
Author(s):  
Oswin Aichholzer ◽  
Franz Aurenhammer ◽  
Thomas Hackl ◽  
Clemens Huemer ◽  
Alexander Pilz ◽  
...  
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Plane Graphs ◽  
Np Completeness ◽  
Complete Problem ◽  
Polynomial Time Algorithms ◽  
Complexity Results ◽  
The Family ◽  
General Plane ◽  
Np Complete

Deciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudo-triangulations, which are a generalization of triangulations, and prove NP-completeness for this class. This result also holds if we bound their face degree to four, or exclusively consider pointed pseudo-triangulations with maximum face degree five. In contrast to these completeness results, we show that pointed pseudo-triangulations with maximum face degree four are always 3-colorable. An according 3-coloring can be found in linear time. Some complexity results relating to the rank of pseudo-triangulations are also given.

scholarly journals Combinatorics of one-dimensional simple Toeplitz subshifts

2018 ◽  
Vol 40 (6) ◽  
pp. 1673-1714
Author(s):  
DANIEL SELL
Keyword(s):  
Systematic Study ◽  
Henri Poincaré ◽  
Explicit Formulas ◽  
De Bruijn Graphs ◽  
One Dimensional ◽  
Combinatorial Properties ◽  
Complexity Function ◽  
Schreier Graphs ◽  
The Family ◽  
De Bruijn

This paper provides a systematic study of fundamental combinatorial properties of one-dimensional, two-sided infinite simple Toeplitz subshifts. Explicit formulas for the complexity function, the palindrome complexity function and the repetitivity function are proved. Moreover, a complete description of the de Bruijn graphs of the subshifts is given. Finally, the Boshernitzan condition is characterized in terms of combinatorial quantities, based on a recent result of Liu and Qu [Uniform convergence of Schrödinger cocycles over simple Toeplitz subshift. Ann. Henri Poincaré12(1) (2011), 153–172]. Particular simple characterizations are provided for simple Toeplitz subshifts that correspond to the orbital Schreier graphs of the family of Grigorchuk’s groups, a class of subshifts that serves as the main example throughout the paper.

scholarly journals New algebraic properties of quadratic quotients of the Rees algebra

2019 ◽  
Vol 18 (03) ◽  
pp. 1950047 ◽  
Author(s):  
Marco D’Anna ◽  
Francesco Strazzanti
Keyword(s):  
Prime Ideal ◽  
Integral Domain ◽  
Rees Algebra ◽  
Ideal Formula ◽  
Local Case ◽  
The Family ◽  
Algebraic Properties

We study some properties of a family of rings [Formula: see text] that are obtained as quotients of the Rees algebra associated with a ring [Formula: see text] and an ideal [Formula: see text]. In particular, we give a complete description of the spectrum of every member of the family and describe the localizations at a prime ideal. Consequently, we are able to characterize the Cohen–Macaulay and Gorenstein properties, generalizing known results stated in the local case. Moreover, we study when [Formula: see text] is an integral domain, reduced, quasi-Gorenstein, or satisfies Serre’s conditions.

Sign in / Sign up

Export Citation Format

Share Document