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Computing Minimal Generating Sets of Invariant Rings of Permutation Groups with SAGBI-Gröbner Basis

2001
*
Discrete Mathematics & Theoretical Computer Science
*

This theory takes,

doi:10.46298/dmtcs.2285
fatcat:zrgiviiqyfclvpthedwsl5aame
*in*this special case, a strongly combinatorial flavor, which makes it particularly*effective*. ... Our implementation, as part*of*the library $permuvar$ for $mupad$, is*in*many cases much more efficient than the other existing software. ... Then, B is a Gröbner basis*of*I if, and only if, for any S-pair s*of**two*invariants*of*B, the remainder*of*s on division by B is zero. This*characterization*is not*effective*. ...##
###
A hypergraph characterization of nearly complete intersections
[article]

2021
*
arXiv
*
pre-print

Stone and Miller then

arXiv:2101.08401v2
fatcat:2aq3p66fnverffrdtnxdkrgp2e
*characterized*nearly*complete*intersections using the theory*of*edge*ideals*(arXiv:2101.07901). ... Recently, nearly*complete*intersection*ideals*were defined by Boocher and Seiner to establish lower bounds on Betti numbers for*monomial**ideals*(arXiv:1706.09866). ...*In*[6] , Stone and Miller provide a*characterization**of*the graphs*of*these*ideals*using*two*small forbidden subgraphs,*completing*our description*of*NCI-hypergraphs. ...##
###
Term-ordering free involutive bases
[article]

2013
*
arXiv
*
pre-print

Here, we introduce and

arXiv:1310.0916v1
fatcat:tuxa75h2bvazlkuvi5irgxzw34
*characterize*for every*monomial**ideal*J a particular*complete*set*of*generators F(J), called stably*complete*, that allows*an*explicit description*of*the family Mf(J). ...*In*this paper, we consider a*monomial**ideal*J*in*P := A[x1,... ... Problem: Given any*monomial**ideal*J ⊳ P find a*characterization*for the family Mf (J)*of*all homogeneous*ideals*I ⊳ P such that the A-module P/I is free with basis given by the set*of*terms*in*the Gröbner ...##
###
Powers of squarefree monomial ideals and combinatorics
[article]

2013
*
arXiv
*
pre-print

*In*addition, we discuss the equivalence between the Conforti-Cornuejols conjecture from linear programming and the question

*of*when symbolic and ordinary powers

*of*squarefree

*monomial*

*ideals*coincide. ... We survey research relating algebraic properties

*of*powers

*of*squarefree

*monomial*

*ideals*to combinatorial structures. ... The

*ideal*membership problems

*in*Theorems 3.7 and 3.9 are for

*monomial*

*ideals*, and so they are computationally simple. On the other hand, computing the chromatic number is

*an*NP-

*complete*problem. ...

##
###
Term-ordering free involutive bases

2015
*
Journal of symbolic computation
*

Here, we introduce and

doi:10.1016/j.jsc.2014.09.005
fatcat:hjea7jhor5cz3od2gybp7ihkeu
*characterize*for every*monomial**ideal*J a particular*complete*set*of*generators F(J), called stably*complete*, that allows*an*explicit description*of*the family Mf (J). ...*In*this paper, we consider a*monomial**ideal*J P := A[x 1 , . . . , x n ], over a commutative ring A, and we face the problem*of*the*characterization*for the family Mf (J)*of*all homogeneous*ideals*I P ... Problem: Given any*monomial**ideal*J P find a*characterization*for the family Mf (J)*of*all homogeneous*ideals*I P such that the A-module P/I is free with basis given by the set*of*terms*in*the Gröbner ...##
###
Invariants of some classes of monomial ideals
[article]

2011
*
arXiv
*
pre-print

*In*this thesis we are interested

*in*describing some homological invariants

*of*certain classes

*of*

*monomial*

*ideals*. We will pay attention to the squarefree and non-squarefree lexsegment

*ideals*. ...

*In*the case

*of*squarefree

*monomial*

*ideals*, the irreducible squarefree

*monomial*

*ideals*are generated by subsets

*of*

*variables*. ...

*In*other words, the irreducible

*monomial*

*ideals*are generated by pure powers

*of*

*variables*. ...

##
###
Involutive Algorithms for Computing Groebner Bases
[article]

2005
*
arXiv
*
pre-print

*In*this paper we describe

*an*efficient involutive algorithm for constructing Groebner bases

*of*polynomial

*ideals*. ... The algorithm is based on the concept

*of*involutive

*monomial*division which restricts the conventional division

*in*a certain way. ... The fundamental property

*of*a Gröbner basis

*of*a polynomial

*ideal*is the divisibility

*of*any element

*in*the related initial

*ideal*by the leading

*monomial*

*of*

*an*element

*in*the basis. ...

##
###
Multigraded Hilbert Series of noncommutative modules
[article]

2018
*
arXiv
*
pre-print

Finally, we present

arXiv:1705.01083v4
fatcat:3qwcasj7hzhn3iz6a2mlzzqdn4
*an*efficient and*complete*implementation*of*(standard) graded and multigraded Hilbert series that has been developed*in*the kernel*of*the computer algebra system Singular. ... Moreover, a*characterization**of*finite-dimensional algebras is obtained*in*terms*of*the nilpotency*of*a key matrix involved*in*the computations. ...*In*particular, we want to thank Hans Schönemann for his cooperation on the implementation*of*the algorithms. ...##
###
On the complexity of radicals in noncommutative rings

2009
*
Journal of Algebra
*

Π 1 1 -

doi:10.1016/j.jalgebra.2009.07.039
fatcat:pjsaqzwls5a5xiwa67owzcfakq
*complete*, while the Levitzki radical*of*the other is Π 0 2 -*complete*. ... This article expands upon the recent work by Downey et al. (2007) [3], who classified the complexity*of*the nilradical and Jacobson radical*of*commutative rings*in*terms*of*the arithmetical hierarchy. ... By*ideal*we mean*two*-sided*ideal*. Definition 1. 2.*An**ideal*P ⊂ R is prime if whenever A B ⊆ P , for*ideals*A, B ⊆ R then either A ⊆ P , or else B ⊆ P . ...##
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An Algebraic Approach for the Unsatisfiability of Nonlinear Constraints
[chapter]

2005
*
Lecture Notes in Computer Science
*

The procedure consists

doi:10.1007/11538363_18
fatcat:vamile5serbn7kjayrh34bus3e
*of*Gröbner basis computation plus extension rules that introduce new definitions, and hence it can be described as a critical-pair*completion*-based logical procedure. ... Such terminating, but potentially incomplete, procedures are used*in*"incompleteness-tolerant" applications. ...*Completeness*Relative to*an*Oracle If*an*oracle can identify the*monomials*p 2 i ν i that are present*in*the witness, then the introduction*of*definitions and computing Gröbner basis is a sound and*complete*...##
###
Maurice Janet's algorithms on systems of linear partial differential equations
[article]

2020
*
arXiv
*
pre-print

His constructions implicitly used

arXiv:1911.09703v2
fatcat:2xg4dyjgb5flpeywu4dkizzxua
*an*interpretation*of*a*monomial*PDE system as a generating family*of*a multiplicative set*of**monomials*. ... Janet was a pioneer*in*the development*of*these algorithmic methods, and the*completion*procedure that he introduced on polynomials was the first one*in*a long and rich series*of*works on*completion*methods ... I Given*an**ideal*generated by a set*of**monomials*, Janet distinguished the family*of**monomials*contained*in*the*ideal*and those contained*in*the complement*of*the*ideal*. ...##
###
Powers of Monomial Ideals and Combinatorics
[article]

2018
*
arXiv
*
pre-print

This is

arXiv:1809.07464v1
fatcat:jk4bblyvbvaddopwv2md3yelsq
*an*exposition*of*some new results on associated primes and the depth*of*different kinds*of*powers*of**monomial**ideals**in*order to show a deep connection between commutative algebra and some objects ...*in*combinatorics such as simplicial complexes, integral points*in*polytopes and graphs. ... Shum for inviting me to be a keynote speaker at the Third International Congress*in*Algebras and Combinatorics (ICAC 2017), Hong Kong, where I had chance to give this lecture to a broad audience. ...##
###
Depth functions and symbolic depth functions of homogeneous ideals
[article]

2021
*
arXiv
*
pre-print

We survey recent studies and results on the following problem: which numerical functions can be the depth function

arXiv:2106.08203v1
fatcat:zoex2lmt2fgpdlq35igwdmwdhi
*of*powers and symbolic powers*of*homogeneous*ideals*. ... However, a modification*of*this approach has finally led to a*complete**characterization**of*depth functions*of**monomial**ideals*, which confirms Conjecture 2.3. ... This*completes*the construction*of**monomial**ideals*with depth functions*of*Type II and, therefore, the proof*of*Theorem 2.5. ...##
###
Linear strands of edge ideals of multipartite uniform clutters
[article]

2020
*
arXiv
*
pre-print

*In*the case that the edge

*ideals*

*of*such clutters have linear resolutions, we give

*an*explicit and surprisingly simple description

*of*their minimal free resolutions, generalizing the known resolutions ... Furthermore, we restate a

*characterization*for edge

*ideals*

*of*d-partite d-uniform clutters which have linear resolutions based on the recent

*characterization*

*of*arithmetically Cohen-Macaulay sets

*of*points ... This research was supported by a grant from the Institute for Research

*in*Fundamental Sciences (IPM), Tehran, Iran. ...

##
###
Neural Codes and the Factor Complex
[article]

2019
*
arXiv
*
pre-print

We use these results to obtain algebraic and combinatorial

arXiv:1904.03235v2
fatcat:yhslixyykvd4nl45adtkbdyifa
*characterizations**of*max-intersection-*complete*codes, as well as a new combinatorial*characterization**of*intersection-*complete*codes. ... We introduce the factor complex*of*a neural code, and show how intervals and maximal codewords are captured by the combinatorics*of*factor complexes. ... Note that the polarization*of*a pseudomonomial*ideal*is a squarefree*monomial**ideal**in*S, that is,*an**ideal*generated by*monomials*that are not divisible by the squares*of*the*variables*(so, P(J) is radical ...
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