@comment -*-texinfo-*-
@comment $Id: plumath.doc,v 1.21 2005-03-08 13:50:34 levandov Exp $
@comment this file contains the mathematical background of Singular:Plural

@c The following directives are necessary for proper compilation
@c with emacs (C-c C-e C-r).  Please keep it as it is.  Since it
@c is wrapped in `@ignore' and `@end ignore' it does not harm `tex' or
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@c ignores the `@ignore').
@ignore
%**start
\input texinfo.tex
@setfilename plumath.info

@c @menu
@c * G-algebras::
@c * Groebner bases @value{PSUFFIX}::  
@c * Hilbert function @value{PSUFFIX}::  
@c * Syzygies and resolutions @value{PSUFFIX}::  
@c * References @value{PSUFFIX}::   
@c @end menu

@node Top, Mathematical background @value{PSUFFIX},(dir), (dir)
@menu
* Mathematical background @value{PSUFFIX}::
@end menu

@ifset singularmanual
@node Mathematical background @value{PSUFFIX},,Functions and system variables @value{PSUFFIX},PLURAL 
@end ifset

@ifclear singularmanual
@node Mathematical background @value{PSUFFIX}, G-algebras, PLURAL, PLURAL
@end ifclear

@section Mathematical background @value{PSUFFIX}
%**end
@end ignore 


This chapter introduces some of the mathematical notions and definitions used
throughout the @sc{Plural} manual. For details, please, refer to
some articles or text books (see @ref{References @value{PSUFFIX}}).

@menu
* G-algebras::
* Groebner bases::
* Syzygies and resolutions @value{PSUFFIX}::
* References @value{PSUFFIX}::
@end menu




@c ---------------------------------------------------------------------------
@node G-algebras, Groebner bases, ,Mathematical background @value{PSUFFIX} 
@section G-algebras
@cindex G-algebra


@subheading Definition

@tex
 Let $K$ be a field, and let a $K$-algebra $A$ be given in terms of generators and relations:
$A= K \langle x_1, \ldots ,x_n \mid$ 
$\{ x_j x_i=c_{ij} \cdot x_i x_j + d_{ij}\}, 1 \leq i <j \leq n \rangle$,
where $c_{ij} \in K^{*}, d_{ij} \in K[x_1, \ldots, x_n]$.
 $A$ is called \textbf{a $G$--algebra}, if the following conditions hold:
@end tex

@ifinfo
 Let K be a field, and let a K-algebra A be given in terms of generators and relations:
A= K < x_1,... ,x_n | 
@{x_j x_i=c_@{ij@} ..., x_i x_j + d_@{ij@} @}, 0< i <j < n+1 >,
where c_@{ij@} in K^*, d_@{ij@} in K[x_1, ..., x_n].
 A is called a G--algebra, if the following conditions hold:
@end ifinfo


@c table @asis 
@itemize @bullet
@item 
@tex
there is a monomial well-ordering $<_A$ such that 
                        $\forall i<j \; \hbox{lm}(d_{ij})<_A x_i x_j$,
@end tex                        


@ifinfo
there is a monomial well-ordering <_A such that 
                        for all i<j lm(d_@{ij@})<_A x_i x_j,
@end ifinfo 


@item @strong{non-degeneracy conditions}: 
@tex
$\forall \; 1 \leq i<j<k \leq n \; \; : \; {\cal NDC}_{ijk} =0$, where
$$
{\cal NDC}_{ijk} = c_{ik}c_{jk} \cdot d_{ij}x_k - x_k d_{ij} + 
c_{jk} \cdot x_j d_{ik} - c_{ij} \cdot d_{ik}x_j + d_{jk}x_i 
- c_{ij}c_{ik} \cdot x_i d_{jk}.
$$
@end tex

@ifinfo
For all  0< i<j<k< n+1  NDC_@{ijk@} =0, where
NDC_@{ijk@}= c_@{ik@}  c_@{jk@} d_@{ij@} x_k - x_k d_@{ij@} + 
c_@{jk@} x_j d_@{ik@} - c_@{ij@} d_@{ik@} x_j + d_@{jk@}x_i 
- c_@{ij@}c_@{ik@} x_i d_@{jk@}.

@end ifinfo

@end itemize

@c @end table


@subheading Theorem
@c @table @asis

@tex
Let $A$ be a $G$-algebra. Then
@end tex

@ifinfo
Let A be a G-algebra. Then
@end ifinfo

@itemize 
@item 
@tex 
$A$ has a PBW (Poincar\'e-Birkhoff-Witt) basis,
@end tex

@ifinfo
 A has a PBW (Poincar'e-Birkhoff-Witt) basis,
@end ifinfo

@item  
@tex 
$A$ 
is left and right noetherian,  
@end tex

@item 
@tex  
$A$ is an integral domain,
@end tex

@ifinfo  
A is an integral domain,
@end  ifinfo


@item  
@tex  
$A$ has left and right quotient rings.
@end tex

@ifinfo  
A has a left and right quotient ring.
@end ifinfo

@end itemize

@*

@tex
In order to set up a $G$--algebra 
in {\sc Plural}, one has to do the following steps:
@end tex

@ifinfo
In order to set up a G-algebra 
in @sc{Plural}, one has to do the following steps:
@end ifinfo

@itemize 

@item 
@tex
 define a commutative ring $R= K[x_1, \ldots, x_n]$, equipped 
 with a global monomial ordering,
@end tex

@ifinfo
 define a commutative ring R= K[x_1, ..., x_n], equipped 
 with a global monomial ordering,
@end ifinfo

@item  
define strictly
@tex 
$n\times n$ upper triangular matrices 
@end tex
@ifinfo
n x n upper triangular matrices 
@end ifinfo
 

@enumerate  
@c @bullet 
@c @table @asis

@item 
@tex 
 $C=\{c_{ij}, i<j\}$, with nonzero entries of type number,
@end tex
@ifinfo 
 C=@{c_@{ij@}, i<j @}, with nonzero entries of type number,
@end ifinfo

@item
@tex
  $D=\{d_{ij}, i<j\}$, with polynomial entries from $R$.
@end tex

@ifinfo
  D=@{d_@{ij@}, i<j@}, with polynomial entries from R.
@end ifinfo


@end enumerate
@c @end table


@item 
call the initialization function @code{ncalgebra} (@ref{ncalgebra})
with the data
@tex
$C$ and $D$.
@end tex
@ifinfo
C and D.
@end ifinfo
 

@end itemize

At present, we do not check automatically whether non-degeneracy conditions
hold but provide a procedure @code{ndc} from the library @code{nctools.lib} instead.


@c ---------------------------------------------------------------------------
@node Groebner bases, Syzygies and resolutions @value{PSUFFIX}, G-algebras, Mathematical background @value{PSUFFIX} 
@section Groebner bases @value{PSUFFIX}
@cindex Groebner bases @value{PSUFFIX}


@tex
Let $<$ be a fixed monomial well-ordering on the $G$--algebra $A$.
@end tex

@ifinfo
Let < be a fixed monomial well-ordering on the G-algebra A.
@end ifinfo


@subheading Definition
@tex
For a set $S \subset A^r$, define 
$\ell(S) \subseteq N^n$ to be the monoid, generated by the leading exponents
of elements of $S$, that is 
$\ell(S)=\langle \alpha \mid \exists s \in S, \hbox{lm(s)}=x^{\alpha} \rangle
\subseteq N^n$. We call $\ell(S)$ the \textbf{monoid of leading exponents}.
There exist $\alpha_1, \ldots, \alpha_m \in N^n$,
such that $\ell(S) :=\langle \alpha_1, \ldots, \alpha_m \rangle$.
We define a \textbf{set of leading monomials of $S$} be
$L(S) := \{ x^{\alpha} \mid \alpha \in \ell(S) \}\subset{A}$.
@end tex

@ifinfo
For a set S a subset A^r, define 
l(S) to be the monoid, generated by the leading exponents
of elements of S, that is 
l(S)=<a| exists s in S, lm(s)=x^a>
is subset N^n. We call l(S) the monoid of leading exponents.
There exist a_1, ..., a_m in N^n,
such that l(S) :=< a_1,..., a_m>.
We define a @strong{set of leading monomials of S} be
L(S) := @{ x^a| a in l(S) @} subset A.
@end ifinfo


@c Let $ I\subset A^r $ be a submodule of $A^r$.
@c Denote by $L(I)$ the submodule of $A^r$ generated by the leading terms 
@c of elements of $I$, i.e. by $\left\{\hbox{lm(f)} \mid f \in I\right\}$.

@tex
A finite set $G\subset I$ is called  {\bf Groebner basis} of $I$ if and
only if $L(G)=L(I)$, that is for any $f \in I\setminus \{ 0 \}$ there exists a $g\in G$
satisfying $ \hbox{lm}(g) \mid \hbox{lm}(f)$.
@end tex
@ifinfo
A finite set G subset I is called  @strong{ Groebner basis} of I if and
only if L(G)=L(I), that is for any f in I\ @{ 0 @} there exists a g in G
satisfying lm(g)|lm(f).
@end ifinfo


@*@strong{Remark:} In the non-commutative case we are working with well
ordering only. (See @ref{PLURAL conventions})

@tex
A Groebner basis $G\subset A^r$ is called {\bf minimal} if $0\notin G$ and if
lm$(g)\notin L(G\setminus \{ g \})$ for all $g\in G$.  
Note, that any Groebner basis can be made minimal by deleting successively those
$g$ with $\hbox{lm}(h)\mid \hbox{lm}(g)$ for some $h\in G\setminus\{g \}$.

For $f\in A^r $ and $ G\subset A^r $ we say that $f$ is {\bf reduced with
respect to $G$} if no monomial of $f$ is contained in $L(G)$.
@end tex

@ifinfo
A Groebner basis G subset A^r is called @strong{minimal} if 0 not in G and if
lm(g) not in L(G\@{ g @}) for all g in G.  
Note, that any Groebner basis can be made minimal by deleting successively those
g with lm(h)| lm(g) for some h in G\@{g@}.

For f in A^r  and  G subset A^r  we say that f  is @strong{reduced with
respect to G} if no monomial of f is contained in L(G).
@end ifinfo


@subheading Normal Form
@table @asis
@tex
A map $\hbox{NF} : A^r \times \{G \mid G\ \hbox{ a Groebner
basis}\} \to A^r, (f|G) \mapsto \hbox{NF}(f|G)$, is called a {\bf normal
form} on $A^r$ if for any $f \in A^r$ and any  Groebner basis $G$ the following
holds: 

(i) $\hbox{NF}(f|G) \not= 0$ then $\hbox{lm}(g)$  does not divide
$\hbox{lm}(\hbox{NF}(f|G))$ for all $g \in G$.

(ii) $f - \hbox{NF}(f|G)\in \langle G \rangle.$ 


\noindent
$\hbox{NF}(f|G)$ is called a {\bf normal form of} $f$ {\bf with
respect to} $G$ (note that such a map is not unique).
@end tex

@ifinfo
A map NF : A^r x G| G\ @{ a Groebner basis@} --> A^r, 

(f|G) --> NF(f|G), is called a @strong{normal form} on A^r if for any f in A^r and any  Groebner basis G the following
holds: 

(i) NF(f|G)<> 0 then lm(g)  does not divide
lm(NF(f|G)) for all g \in G.

(ii) f - NF(f|G) is in <G>.$ 


NF(f|G) is called a @strong{normal form of} f @strong{with respect to}
 G (note that such a map is not unique).
@end ifinfo
@end table

@*@strong{ Remark:}
With respect to the definitions of @code{ideal} and @code{module} (see
@ref{PLURAL conventions} )  @sc{Plural} works with left normal form only.



@table @asis
@item Ideal membership:
@cindex Ideal membership
@tex
For a Groebner basis $G$ of $I$ the following holds: 
$f \in I$ if and only if $\hbox{NF}(f|G) = 0$.
@end tex

@ifinfo
For a Groebner basis G of I the following holds: 
f is in I if and only if NF(f|G) = 0.
@end ifinfo
@end table

@c @item Hilbert function:
@c @ifset singularmanual 
@c see @ref{Hilbert function}
@c @end ifset 
 
@c @ifclear singularmanual
@c See section Hilbert function in chapter
@c Mathematical background @sc{Singular}
@c manual.
@c @end ifclear
@c @end table 





@c ---------------------------------------------------------------------------


@c @node Hilbert function @value{PSUFFIX}, Syzygies and resolutions @value{PSUFFIX}, Groebner bases @value{PSUFFIX}, Mathematical background @value{PSUFFIX} 
@c @section Hilbert function @value{PSUFFIX}
@c @cindex Hilbert function @value{PSUFFIX}
@c @cindex Hilbert series @value{PSUFFIX}
@c @tex
@c Let M $=\bigoplus_i M_i$ be a graded module over $K[x_1,..,x_n]$ with 
@c respect to weights $(w_1,..w_n)$.
@c The {\bf Hilbert function} of $M$, $H_M$, is defined (on the integers) by
@c $$H_M(k) :=dim_K M_k.$$
@c The {\bf Hilbert-Poincare series}  of $M$ is the power series
@c $$\hbox{HP}_M(t) :=\sum_{i=-\infty}^\infty
@c H_M(i)t^i=\sum_{i=-\infty}^\infty dim_K M_i \cdot t^i.$$
@c It turns out that $\hbox{HP}_M(t)$ can be written in two useful ways
@c for weights $(1,..,1)$:
@c $$\hbox{HP}_M(t)={Q(t)\over (1-t)^n}={P(t)\over (1-t)^{dim(M)}}$$
@c where $Q(t)$ and $P(t)$ are polynomials in ${\bf Z}[t]$.
@c $Q(t)$ is called the {\bf first Hilbert series},
@c and $P(t)$ the {\bf second Hilbert series}.
@c If \hbox{$P(t)=\sum_{k=0}^N a_k t^k$}, and \hbox{$d = dim(M)$},
@c then \hbox{$H_M(s)=\sum_{k=0}^N a_k$ ${d+s-k-1}\choose{d-1}$}
@c (the {\bf Hilbert polynomial}) for $s \ge N$.
@c @end tex
@c @ifinfo
@c Let M =(+) M_i be a graded module over K[x_1,...,x_n] with
@c respect to weights (w_1,..w_n).
@c The Hilbert function of M H_M is defined by
@c @display
@c H_M(k)=dim_K M_k.
@c @end display
@c The Hilbert-Poincare series  of M is the power series
@c @display
@c HP_M(t)=sum_i dim_K (M_i)*t^i.
@c @end display
@c It turns out that HP_M(t) can be written in two useful ways
@c for weights $(1,..,1)$:
@c @display
@c H_M(t)=Q(t)/(1-t)^n=P(t)/(1-t)^dim(M).
@c @end display
@c where Q(t) and P(t) are polynomials in Z[t].
@c Q(t) is called the first Hilbert series, and P(t) the second Hilbert series.
@c If P(t)=sum_(k=0)^N a_k t^k, and d=dim(M),
@c then
@c @display
@c H_M(s)=sum_(k=0)^N a_k binomial(d+s-k-1,d-1) (the Hilbert polynomial)
@c @end display
@c for s >= N.
@c @end ifinfo
@c @*
@c @*
@c @tex
@c Generalizing these to quasihomogeneous modules we get
@c $$\hbox{HP}_M(t)={Q(t)\over {\Pi_{i=1}^n(1-t^{w_i})}}$$
@c where $Q(t)$ is a polynomial in ${\bf Z}[t]$.
@c $Q(t)$ is called the {\bf first (weighted) Hilbert series} of M.
@c @end tex
@c @ifinfo
@c Generalizing these to quasihomogeneous modules we get
@c @display
@c H_M(t)=Q(t)/Prod((1-t)^(w_i)).
@c @end display
@c where Q(t) is a polynomial in Z[t].
@c Q(t) is called the first (weighted) Hilbert series of M.
@c @end ifinfo

@c ---------------------------------------------------------------------------
@node Syzygies and resolutions @value{PSUFFIX}, References @value{PSUFFIX}, Groebner bases, Mathematical background @value{PSUFFIX} 
@section Syzygies and resolutions @value{PSUFFIX}
@cindex Syzygies and resolutions @value{PSUFFIX}

@subheading Syzygies
@tex
Let $K$ be a field and $>$ a well ordering on $A^r$. A {\bf left syzygy}
between $k$ elements $f_1,\dots,f_k \in A^r =\oplus_{i=1}^{r}Ae_i $ is
$k$-tuple $(g_1,\dots ,g_k)\in A^k$ satisfying 
$$\sum_{i=1}^{k}g_i f_i =0  $$
The set of all left (resp.right)  syzygies between $f_1,...,f_k$ is left (resp.
right) submodule $S$ of $A^k$. 
@end tex

@ifinfo
Let K be a field and > a well ordering on A^r. A @strong{left syzygy}
between k elements f_1,...,f_k in A^r is
k-tuple (g_1,... ,g_k) in A^k satisfying 
 sum_@{i=1@}^@{k@}g_i f_i =0  
The set of all left (resp.right)  syzygies between f_1,...,f_k is left (resp.
right) submodule S of A^k. 
@end ifinfo



@*@strong{Remark:}
With respect to the definitions of @code{ideal} and  @code{module}  (see @ref{PLURAL conventions} )  @sc{Plural} works with left
syzygies only (by @code{syz} we understand  a  left syzygy). 
If @code{S} is a matrix of a left syzygy module of left submodule given by matrix @code{M}, then @code{transpose(S)*transpose(M) = 0}.


@tex
Note, that the syzygy modules of $I$ depend on a choice of generators
$g_1, \dots , g_s$.
But one can show that they depend on $I$ uniquely up to direct summands.
@end tex
@ifinfo
Note, that the syzygy modules of I depend on a choice of generators
g_1,\dots , g_s.
But one can show that they depend on I uniquely up to direct summands.
@end ifinfo


@subheading Free resolutions
@tex
Let $I=\langle g_1,\dots ,g_s\rangle \subseteq A^r$ and $M= A^r/I$.
A {\bf free resolution of $M$} is a long exact sequence
$$\dots \longrightarrow F_2 \buildrel{B_2}\over{\longrightarrow} F_1
 \buildrel{B_1}\over{\longrightarrow} F_0\longrightarrow M\longrightarrow
0,$$
@end tex
@ifinfo
Let I=<g_1,... ,g_s> in A^r and M=A^r/I.  A free resolution of M is a
long exact sequence
@display
...--> F2 --A2-> F1 --A1-> F0-->M-->0,
@end display
@end ifinfo
@*where the columns of the matrix
@tex
$B_1$
@end tex
@ifinfo
B_1
@end ifinfo
generate @math{I}. Note, that resolutions over factor-algebras need not to be finite (i.e., of finite length). 
The Generalized Hilbert Syzygy Theorem states, that for G-algebra 
@tex
$A$,
@end tex
@ifinfo
A,
@end ifinfo
 generated by n variables, there exists a  resolution of length smaller or equal than n.

@table @code
@item @strong{Example:}
@smallexample
@c example
LIB "poly.lib";
ring R=0,(x,y,z),dp;
matrix d[3][3];
d[1,2]=-z;  d[1,3]=2x;  d[2,3]=-2y;
ncalgebra(1,d); // this is U(sl_2)
ideal I=x3,y3,z3-z;
I=std(I);
I;
resolution resI = mres(I,0); 
resI;
// The matrix A_1 is given by
print(matrix(resI[1]));
// We see that the columns of A_1 generate I.
// The matrix A_2 is given by
print(matrix(resI[2]));
ideal tst; // now let us show that the resolution is exact
matrix TST;
TST = transpose(resI[3])*transpose(resI[2]); // 2nd term
tst = std(flatten(TST));
tst;
TST = transpose(resI[2])*transpose(resI[1]); // 1st term
tst = std(flatten(TST));
tst;

@c example
@end smallexample
@end table

@c ---------------------------------------------------------------------------
@node References @value{PSUFFIX}, , Syzygies and resolutions @value{PSUFFIX}, Mathematical background @value{PSUFFIX}
@section References @value{PSUFFIX}
@cindex References @value{PSUFFIX}

The Centre for Computer Algebra Kaiserslautern publishes a series of preprints
which are electronically available at
@code{http://www.mathematik.uni-kl.de/~zca/Reports_on_ca}.
Other sources to check are the following books and articles:

@subheading Text books

@c @include plumatref.tex

@itemize @bullet
@item
@c DK  book
Y. Drozd and V. Kirichenko.
    Finite dimensional algebras. With an appendix by Vlastimil
  Dlab.
   Springer, 1994

@item
@c GPS  book
 [GPS] Greuel, G.-M. and Pfister, G. with contributions by Bachmann, O. ; Lossen, C.
  and Sch@"onemann, H. 
     A SINGULAR Introduction to Commutative Algebra.
   Springer, 2002

@item 
@c BGV
 [BGV] Bueso, J.; Gomez Torrecillas, J.; Verschoren, A.
   Algorithmic methods in non-commutative algebra. Applications to quantum groups.
   Kluwer Academic Publishers, 2003

@item
@c Kr  Book
 Kredel, H. 
     Solvable polynomial rings.
  Shaker, 1993

@item
@c HLi  Book
   [Li] Huishi Li.
     Noncommutative Gr@"obner bases and filtered-graded transfer.
   Springer, 2002

@item
@c MR  book
 [MR] McConnell, J.C. and Robson, J.C. 
Noncommutative Noetherian rings. With the cooperation of L. W. Small.
Graduate Studies in Mathematics. 30. Providence, RI: American Mathematical Society (AMS).,
2001

@end itemize

@subheading Descriptions of algorithms and problems

@itemize @bullet

@item
@c{HKP} art
  Havlicek, M. and Klimyk, A. and Posta, S. 
   Central elements of the algebras
@tex
 $U'_q({\rm so}_m)$ and $U'_q({\rm iso}_m)$.
  {arXiv. math. QA/9911130}, (1999)
@end tex

@item
@c{AP} art
J. Apel.
  Gr@"obnerbasen in nichtkommutativen algebren und ihre anwendung.
    Dissertation, Universit@"at Leipzig, 1988.

@item
@c{AP2}  art
 Apel, J. 
  Computational ideal theory in finitely generated extension rings.
    Theor. Comput. Sci.(2000), 244(1-2):1--33

@item
@c{BachS:98} InCollection
O. Bachmann and H. Sch@"onemann.
   Monomial operations for computations of Gr@"obner bases.
  In   Reports On Computer Algebra 18. Centre for Computer Algebra, 
University of Kaiserslautern (1998)
 

@item
@c DE} InProceedings
D. Decker and D. Eisenbud.
  Sheaf algorithms using the exterior algebra.
  In  Eisenbud, D.; Grayson, D.; Stillman, M.; Sturmfels, B., editor,
     Computations in algebraic geometry with Macaulay 2, (2001)

@item
@c art
Jose L. Bueso, J. Gomez Torrecillas and F. J. Lobillo. 
  Computing the Gelfand-Kirillov dimension II. 
    In  A. Granja, J. A. Hermida and A. Verschoren eds. Ring Theory and Algebraic Geometry, Lect. Not. in Pure and Appl. Maths., Marcel Dekker, 2001.

@item
@c art
Jose L. Bueso, J. Gomez Torrecillas and F. J. Lobillo. 
  Re-filtering and exactness of the Gelfand-Kirillov dimension. 
    Bulletin des Sciences Mathematiques 125(8), 689-715 (2001).


@item
@c GL  art
J. Gomez Torrecillas and F.J. Lobillo.
  Global homological dimension of multifiltered rings and quantized
  enveloping algebras.
    J. Algebra, 225(2):522--533, 2000.


@item
@c I1  InProc
N. Iorgov.
@tex  
 On the Center of $q$-Deformed Algebra $U'_q( \rm so _3)$ Related to
  Quantum Gravity at $q$ a Root of $1$.
@end tex  
In   Proceedings of IV Int. Conf. "Symmetry in Nonlinear
  Mathematical Physics",(2001) Kyiv, Ukraine 

@item
@c KW  art
A. Kandri-Rody and V. Weispfenning.
  Non--commutative Gr@"obner bases in algebras of solvable type.
    J. Symbolic Computation, 9(1):1--26, 1990.


@item
@c LV2  Inproc
 Levandovskyy, V. 
   On Gr@"obner bases for non--commutative G-algebras.
  In  Kredel, H. and Seiler, W.K., editor,    Proceedings of the
  8th Rhine Workshop on Computer Algebra, 2002.

@item
@c NDC  InProc
 Levandovskyy, V. 
   PBW Bases, Non--Degeneracy Conditions and Applications.
  In  Buchweitz, R.-O. and Lenzing, H., editor,    Proceedings of
  the ICRA X conference, Toronto, 2003.

@item
@c LS  InProc
Levandovskyy V.; Sch@"onemann, H. 
  Plural --- a computer algebra system for noncommutative polynomial
  algebras.
  In  Proc. of the International Symposium on Symbolic and
  Algebraic Computation (ISSAC'03). ACM Press, 2003.

@item
@c MoraNC  article
 Mora, T. 
   Gr@"obner bases for non-commutative polynomial rings.
     Proc. AAECC 3 Lect. N. Comp. Sci, 229:  353--362, 1986.

@item
@c Mora  article
 Mora, T. 
   An introduction to commutative and non-commutative Groebner bases.
     Theor. Comp. Sci., 134:  131--173, 1994.

@item
@c NS  art
T. N@"u@ss{}ler and H. Sch@"onemann.
  Gr@"obner bases in algebras with zero--divisors.
     Preprint 244, Universit@"at Kaiserslautern, 1993.

@item
@c CMR  art
Ringel, C. M. 
  PBW-bases of quantum groups.
   J. Reine Angew. Math., 470:51--88, 1996.

@item
@c S:03  InColl
Sch@"onemann, H. 
  Singular in a Framework for Polynomial Computations.
  In Joswig, M. and Takayama, N., editor, Algebra, Geometry and
  Software Systems, pages 163--176. Springer, 2003.

@item
@c Yan:98  art
T. Yan.
  The geobucket data structure for polynomials.
   J. Symbolic Computation, 25(3):285--294, March 1998.
@end itemize







@c @item
@c Adams, W.; Loustaunau, P.: An Introduction to Gr@"obner Bases. Providence, RI,
@c AMS, 1996

@c @item
@c Becker, T.; Weisspfenning, V.:
@c Gr@"obner Bases - A Computational Approach to Commutative Algebra. Springer, 1993

@c @item
@c Cohen, H.:
@c A Course in Computational Algebraic Number Theory,
@c Springer, 1995

@c @item
@c Cox, D.; Little, J.; O'Shea, D.:
@c Ideals, Varieties and Algorithms. Springer, 1996

@c @item
@c Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry.
@c Springer, 1995

@c @item
@c Mishra, B.: Algorithmic Algebra, Texts and Monographs in Computer Science.
@c Springer, 1993
@c @item
@c Sturmfels, B.: Algorithms in Invariant Theory. Springer 1993

@c @item
@c Vasconcelos, W.: Computational Methods in Commutative Algebra and Algebraic
@c Geometry. Springer, 1998
@c @end itemize

@c @subheading Descriptions of algorithms
@c @itemize @bullet
@c @item
@c Bareiss, E.:
@c Sylvester's identity and multistep integer-preserving Gaussian elimination.
@c Math. Comp. 22 (1968), 565-578

@c @item
@c Campillo, A.: Algebroid curves in positive characteristic. SLN 813, 1980

@c @item
@c Chou, S.:
@c Mechanical Geometry Theorem Proving.
@c D.Reidel Publishing Company, 1988

@c @item
@c Decker, W.; Greuel, G.-M.; Pfister, G.:
@c Primary decomposition: algorithms and
@c comparisons.  Preprint, Univ. Kaiserslautern, 1998.
@c To appear in: Greuel, G.-M.; Matzat, B. H.; Hiss, G. (Eds.),
@c Algorithmic Algebra and Number Theory. Springer Verlag, Heidelberg, 1998

@c @item
@c Decker, W.; Greuel, G.-M.; de Jong, T.; Pfister, G.:
@c The normalisation: a new algorithm,
@c implementation and comparisons. Preprint, Univ. Kaiserslautern, 1998

@c @item
@c Decker, W.; Heydtmann, A.; Schreyer, F. O.: Generating a Noetherian Normalization of
@c the Invariant Ring of a Finite Group, 1997, to appear in Journal of
@c Symbolic Computation

@c @item
@c @tex
@c Faug\`ere,
@c @end tex
@c @ifinfo
@c Faugere,
@c @end ifinfo
@c J. C.; Gianni, P.; Lazard, D.; Mora, T.: Efficient computation
@c of zero-dimensional
@c Gr@"obner bases by change of ordering. Journal of Symbolic Computation, 1989

@c @item
@c Gr@"abe, H.-G.: On factorized Gr@"obner bases, Univ. Leipzig, Inst. f@"ur
@c Informatik, 1994

@c @item
@c Grassmann, H.; Greuel, G.-M.; Martin, B.; Neumann,
@c W.; Pfister, G.; Pohl, W.; Sch@"onemann, H.; Siebert, T.:  On an
@c implementation of Groebner bases and syzygies in  @sc{Singular}.
@c Proceedings of the Workshop  Computational Methods in Lie theory in AAECC (1995)

@c @item
@c Greuel, G.-M.; Pfister, G.:
@c Advances and improvements in the theory of Standart bases and
@c syzygies. Arch. d. Math. 63(1995)

@c @item
@c Kemper; Generating Invariant Rings of Finite Groups over Arbitrary
@c Fields. 1996, to appear in Journal of Symbolic Computation

@c @item
@c Kemper and Steel: Some Algorithms in Invariant Theory of Finite Groups. 1997

@c @item
@c Lee, H.R.; Saunders, B.D.: Fraction Free Gaussian Elimination for
@c Sparse Matrices. Journal of Symbolic Computation (1995) 19, 393-402

@c @item
@c Sch@"onemann, H.:
@c Algorithms in @sc{Singular},
@c Reports on Computer Algebra 2(1996), Kaiserslautern

@c @item
@c Siebert, T.:
@c On strategies and implementations for computations of free resolutions.
@c Reports on Computer Algebra 8(1996), Kaiserslautern

@c @item
@c Wang, D.:
@c Characteristic Sets and Zero Structure of Polynomial Sets.
@c Lecture Notes, RISC Linz, 1989
@c @end itemize