# Singular

Library:
Purpose:
Algorithms for quotient algebras in the letterplace case
Authors:
Grischa Studzinski, grischa.studzinski@rwth-aachen.de

Support: Joint projects LE 2697/2-1 and KR 1907/3-1 of the Priority Programme SPP 1489:
'Algorithmische und Experimentelle Methoden in Algebra, Geometrie und Zahlentheorie'
of the German DFG

Overview:
Given the free algebra A = K<x_1,...,x_n> and a (finite) Groebner basis GB = {g_1,..,g_w}, one is interested in the K-dimension and in the explicit K-basis of A/<GB>.
Therefore one is interested in the following data:
- the Ufnarovskij graph induced by GB
- the mistletoes of A/<GB>
- the K-dimension of A/<GB>
- the Hilbert series of A/<GB>

The Ufnarovskij graph is used to determine whether A/<GB> has finite K-dimension. One has to check if the graph contains cycles. For the whole theory we refer to [ufna]. Given a
reduced set of monomials GB one can define the basis tree, whose vertex set V consists of all normal monomials w.r.t. GB. For every two monomials m_1, m_2 in V there is a direct edge from m_1 to m_2, if and only if there exists x_k in {x_1,..,x_n}, such that m_1*x_k = m_2. The set M = {m in V | there is no edge from m to another monomial in V} is called the set of mistletoes. As one can easily see it consists of the endpoints of the graph. Since there is a unique path to every monomial in V the whole graph can be described only from the knowledge of the mistletoes. Note that V corresponds to a basis of A/<GB>, so knowing the mistletoes we know a K-basis. The name mistletoes was given to those points because of these miraculous value and the algorithm is named sickle, because a sickle is the tool to harvest mistletoes. For more details see [studzins]. This package uses the Letterplace format introduced by [lls]. The algebra can either be represented as a Letterplace ring or via integer vectors: Every variable will only be represented by its number, so variable one is represented as 1, variable two as 2 and so on. The monomial x_1*x_3*x_2 for example will be stored as (1,3,2). Multiplication is concatenation. Note that there is no algorithm for computing the normal form needed for our case. Note that the name fpadim.lib is short for dimensions of finite presented algebras.

References:

[ufna] Ufnarovskij: Combinatorical and asymptotic methods in algebra, 1990
[lls] Levandovskyy, La Scala: Letterplace ideals and non-commutative Groebner bases, 2009
[studzins] Studzinski: Dimension computations in non-commutative, associative algebras, Diploma thesis, RWTH Aachen, 2010

Assumptions:
- basering is always a Letterplace ring
- all intvecs correspond to Letterplace monomials
- if you specify a different degree bound d, d <= attrib(basering,uptodeg) should hold.
In the procedures below, 'iv' stands for intvec representation and 'lp' for the letterplace representation of monomials

Procedures:

 7.7.7.0. ivDHilbert computes the K-dimension and the Hilbert series 7.7.7.0. ivDHilbertSickle computes mistletoes, K-dimension and Hilbert series 7.7.7.0. ivDimCheck checks if the K-dimension of A/ is infinite 7.7.7.0. ivHilbert computes the Hilbert series of A/ in intvec format 7.7.7.0. ivKDim computes the K-dimension of A/ in intvec format 7.7.7.0. ivMis2Base computes a K-basis of the factor algebra 7.7.7.0. ivMis2Dim computes the K-dimension of the factor algebra 7.7.7.0. ivOrdMisLex orders a list of intvecs lexicographically 7.7.7.0. ivSickle computes the mistletoes of A/ in intvec format 7.7.7.0. ivSickleHil computes the mistletoes and Hilbert series of A/ 7.7.7.0. ivSickleDim computes the mistletoes and the K-dimension of A/ 7.7.7.0. lpDHilbert computes the K-dimension and Hilbert series of A/ 7.7.7.0. lpDHilbertSickle computes mistletoes, K-dimension and Hilbert series 7.7.7.0. lpHilbert computes the Hilbert series of A/ in lp format 7.7.7.0. lpDimCheck checks if the K-dimension of A/ is infinite 7.7.7.0. lpKDim computes the K-dimension of A/ in lp format 7.7.7.0. lpMis2Base computes a K-basis of the factor algebra 7.7.7.0. lpMis2Dim computes the K-dimension of the factor algebra 7.7.7.0. lpOrdMisLex orders an ideal of lp-monomials lexicographically 7.7.7.0. lpSickle computes the mistletoes of A/ in lp format 7.7.7.0. lpSickleHil computes the mistletoes and Hilbert series of A/ 7.7.7.0. lpSickleDim computes the mistletoes and the K-dimension of A/ 7.7.7.0. sickle can be used to access all lp main procedures 7.7.7.0. ivL2lpI transforms a list of intvecs into an ideal of lp monomials 7.7.7.0. iv2lp transforms an intvec into the corresponding monomial 7.7.7.0. iv2lpList transforms a list of intmats into an ideal of lp monomials 7.7.7.0. iv2lpMat transforms an intmat into an ideal of lp monomials 7.7.7.0. lp2iv transforms a polynomial into the corresponding intvec 7.7.7.0. lp2ivId transforms an ideal into the corresponding list of intmats 7.7.7.0. lpId2ivLi transforms a lp-ideal into the corresponding list of intvecs