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

Nicolas Thiéry
2001 Discrete Mathematics & Theoretical Computer Science  
This theory takes, 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.  ... 
doi:10.46298/dmtcs.2285 fatcat:zrgiviiqyfclvpthedwsl5aame

A hypergraph characterization of nearly complete intersections [article]

Chiara Bondi, Courtney R. Gibbons, Yuye Ke, Spencer Martin, Shrunal Pothagoni, Andrew Stelzer
2021 arXiv   pre-print
Stone and Miller then 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.  ... 
arXiv:2101.08401v2 fatcat:2aq3p66fnverffrdtnxdkrgp2e

Term-ordering free involutive bases [article]

Michela Ceria, Teo Mora, Margherita Roggero
2013 arXiv   pre-print
Here, we introduce and 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  ... 
arXiv:1310.0916v1 fatcat:tuxa75h2bvazlkuvi5irgxzw34

Powers of squarefree monomial ideals and combinatorics [article]

Christopher A. Francisco, Huy Tai Ha, Jeffrey Mermin
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.  ... 
arXiv:1303.6642v1 fatcat:6idvqnv2nrfrfkou6b6yusmkgi

Term-ordering free involutive bases

Michela Ceria, Teo Mora, Margherita Roggero
2015 Journal of symbolic computation  
Here, we introduce and 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  ... 
doi:10.1016/j.jsc.2014.09.005 fatcat:hjea7jhor5cz3od2gybp7ihkeu

Invariants of some classes of monomial ideals [article]

Oana Olteanu
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.  ... 
arXiv:1109.2463v1 fatcat:vog45dkpzrevbkej44w67ptrlu

Involutive Algorithms for Computing Groebner Bases [article]

Vladimir P.Gerdt
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.  ... 
arXiv:math/0501111v1 fatcat:duouehfbundkpf2id2ta6bcmam

Multigraded Hilbert Series of noncommutative modules [article]

Roberto La Scala, Sharwan K. Tiwari
2018 arXiv   pre-print
Finally, we present 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.  ... 
arXiv:1705.01083v4 fatcat:3qwcasj7hzhn3iz6a2mlzzqdn4

On the complexity of radicals in noncommutative rings

Chris J. Conidis
2009 Journal of Algebra  
Π 1 1 -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 .  ... 
doi:10.1016/j.jalgebra.2009.07.039 fatcat:pjsaqzwls5a5xiwa67owzcfakq

An Algebraic Approach for the Unsatisfiability of Nonlinear Constraints [chapter]

Ashish Tiwari
2005 Lecture Notes in Computer Science  
The procedure consists 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  ... 
doi:10.1007/11538363_18 fatcat:vamile5serbn7kjayrh34bus3e

Maurice Janet's algorithms on systems of linear partial differential equations [article]

Kenji Iohara, Philippe Malbos
2020 arXiv   pre-print
His constructions implicitly used 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.  ... 
arXiv:1911.09703v2 fatcat:2xg4dyjgb5flpeywu4dkizzxua

Powers of Monomial Ideals and Combinatorics [article]

Le Tuan Hoa
2018 arXiv   pre-print
This is 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.  ... 
arXiv:1809.07464v1 fatcat:jk4bblyvbvaddopwv2md3yelsq

Depth functions and symbolic depth functions of homogeneous ideals [article]

Huy Tai Ha, Ngo Viet Trung
2021 arXiv   pre-print
We survey recent studies and results on the following problem: which numerical functions can be the depth function 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.  ... 
arXiv:2106.08203v1 fatcat:zoex2lmt2fgpdlq35igwdmwdhi

Linear strands of edge ideals of multipartite uniform clutters [article]

Amin Nematbakhsh
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.  ... 
arXiv:1805.11432v2 fatcat:23t4kiqwijav7pqsrcllw3gs5q

Neural Codes and the Factor Complex [article]

Alexander Ruys de Perez, Laura Felicia Matusevich, Anne Shiu
2019 arXiv   pre-print
We use these results to obtain algebraic and combinatorial 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  ... 
arXiv:1904.03235v2 fatcat:yhslixyykvd4nl45adtkbdyifa
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