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@comment this file contains the mathematical background of Singular
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%**start
\input texinfo.tex
@setfilename plumath.info
@c
@c
@c
@c
@c
@c
@c
@menu
* G-algebras::
* Groebner bases @value{PSUFFIX}::
* Hilbert function @value{PSUFFIX}::
* Syzygies and resolutions @value{PSUFFIX}::
* References @value{PSUFFIX}::
@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 an $K$-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 an 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 hold}:
@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 $G$--algebra
in {\sc Plural}, one has to do the following steps:
@end tex
@ifinfo
In order to set up G--algebra
in @sc{Plural}, one has to do the following steps:
@end ifinfo
@itemize
@item
@tex
define a commutative ring $R= K[x_1, \ldots, x_n]$ with a
well--ordering $<$ on it,
@end tex
@ifinfo
define a commutative ring R= K[x_1, ..., x_n] with a
well--ordering < on it,
@end ifinfo
@item
define strictly
@tex
$n\times n$ upper triangular matrices
@end tex
@ifinfo
nxn upper triangular matrices
@end ifinfo
@enumerate
@c @bullet
@c @table @asis
@item
@tex
$C=\{c_{ij}, i<j\}$, with nonzero entries of the type number,
@end tex
@ifinfo
C=@{c_@{ij@}, i<j @}, with nonzero entries of the 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 @ref{ncalgebra}
@tex
initialization function with the data $R$,$C$ and $D$.
@end tex
@ifinfo
call @ref{ncalgebra} initialization function with the data R,C and D.
@end ifinfo
@end itemize
At present, we do not perform checks of leading monomial condition
and non-degeneracy conditions while setting an algebra.
@c -------------------------------------------------------------------------@node Groebner bases, Syzygies and resolutions @value{PSUFFIX}, Galgebras, 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
$\ell(S)$ to be a 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 $\ell(S)$ a \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 a 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) a 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 $g\in G$.
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.
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 $\hbox{lm}(g)$
$\hbox{lm}(\hbox{NF}(f|G))$ for all $g \in G$.
does not divide
(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 function 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 function 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
@c
@c
@c
@item Hilbert function:
@ifset singularmanual
see @ref{Hilbert function}
@end ifset
@c
@c
@c
@c
@c
@c
@ifclear singularmanual
See section Hilbert function in chapter
Mathematical background @sc{Singular}
manual.
@end ifclear
@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}
@ref{PLURAL conventions} ) @sc{Plural} works with left
syzygies only (by @code{syz} we understand a left syzygy).
@tex
Note, that the syzygy
$g_1, \dots , g_s$.
But one can show that
@end tex
@ifinfo
Note, that the syzygy
g_1,\dots , g_s.
But one can show that
@end ifinfo
(see
modules of $I$ depend on a choice of generators
they depend on $I$ uniquely up to direct summands.
modules of I depend on a choice of generators
they depend on I uniquely up to direct summands.
@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 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
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); //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]));
@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 @code{http://symbolicnet.mcs.kent.edu/},
@code{http://www.can.nl/},... and the following list of books:
@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
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
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
H. Li.
Noncommutative Gr@"obner bases and filtered-graded transfer.
Springer, 2002
@item
@c MR book
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
@c
@c
@c
@item
Cohen, H.:
A Course in Computational Algebraic Number Theory,
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
@c
@c
@c
@item
Chou, S.:
Mechanical Geometry Theorem Proving.
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
@c
@c
@c
@item
Decker, W.; Greuel, G.-M.; de Jong, T.; Pfister, G.:
The normalisation: a new algorithm,
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
@c
@c
@c
@item
Greuel, G.-M.; Pfister, G.:
Advances and improvements in the theory of Standart bases and
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
@c
@c
@c
@item
Sch@"onemann, H.:
Algorithms in @sc{Singular},
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
@c
@c
@c
@c
@item
Wang, D.:
Characteristic Sets and Zero Structure of Polynomial Sets.
Lecture Notes, RISC Linz, 1989
@end itemize