@comment -*-texinfo-*- @comment $Id: math.doc,v 1.78 2006-07-17 14:32:21 Singular Exp $ @comment this file contains the mathematical background of Singular @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 @c `makeinfo' but is a great help in editing this file (emacs @c ignores the `@ignore'). @ignore %**start \input texinfo.tex @setfilename math.info @node Top, Mathematical background @menu * General concepts:: @end menu @node Mathematical background, SINGULAR libraries, Examples, Top @chapter Mathematical background %**end @end ignore This chapter introduces some of the mathematical notions and definitions used throughout the manual. It is mostly a collection of the most prominent definitions and properties. For details, please, refer to some articles or text books (see @ref{References}). @menu * Standard bases:: * Hilbert function:: * Syzygies and resolutions:: * Characteristic sets:: * Gauss-Manin connection:: * Toric ideals and integer programming:: @ifset withplural * Non-commutative algebra :: @end ifset * References:: @end menu @c --------------------------------------------------------------------------- @node Standard bases, Hilbert function, ,Mathematical background @section Standard bases @cindex Standard bases @subheading Definition @tex Let $R = \hbox{Loc}_< K[\underline{x}]$ and let $I$ be a submodule of $R^r$. Note that for r=1 this means that $I$ is an ideal in $R$. Denote by $L(I)$ the submodule of $R^r$ generated by the leading terms of elements of $I$, i.e. by $\left\{L(f) \mid f \in I\right\}$. Then $f_1, \ldots, f_s \in I$ is called a {\bf standard basis} of $I$ if $L(f_1), \ldots, L(f_s)$ generate $L(I)$. @end tex @ifinfo Let R = Loc K[x] and let I be a submodule of $R^r$. Denote by L(I) the submodule of R^r generated by the leading terms of elements in $I$, i.e. by @{ L(f) | f in I@}. Then f_1, @dots{}, f_s in I is called a @strong{standard basis} of I if L(f_1), @dots{}, L(f_s) generate L(I). @end ifinfo @subheading Properties @table @asis @item normal form: @cindex Normal form @tex A function $\hbox{NF} : R^r \times \{G \mid G\ \hbox{ a standard basis}\} \to R^r, (p,G) \mapsto \hbox{NF}(p|G)$, is called a {\bf normal form} if for any $p \in R^r$ and any standard basis $G$ the following holds: if $\hbox{NF}(p|G) \not= 0$ then $L(g)$ does not divide $L(\hbox{NF}(p|G))$ for all $g \in G$. \noindent $\hbox{NF}(p|G)$ is called a {\bf normal form of} $p$ {\bf with respect to} $G$ (note that such a function is not unique). @end tex @ifinfo A function NF : R^r x @{G | G a standard basis@} -> R^r, (p,G) -> NF(p|G), is called a @strong{normal form} if for any p in R^r and any standard basis G the following holds: if NF(p|G) <> 0 then L(g) does not divide L(NF(p|G)) for all g in G. @*NF(p|G) is called a @strong{normal form} of p with respect to G (note that such a function is not unique). @end ifinfo @item ideal membership: @cindex Ideal membership @tex For a standard basis $G$ of $I$ the following holds: $f \in I$ if and only if $\hbox{NF}(f,G) = 0$. @end tex @ifinfo For a standart basis G of I the following holds: f in I if and only if NF(f,G) = 0. @end ifinfo @item Hilbert function: @tex Let \hbox{$I \subseteq K[\underline{x}]^r$} be a homogeneous module, then the Hilbert function $H_I$ of $I$ (see below) and the Hilbert function $H_{L(I)}$ of the leading module $L(I)$ coincide, i.e., $H_I=H_{L(I)}$. @end tex @ifinfo Let I in K[x]^r be a homogeneous ideal, then the Hilbert function H_I of I and the Hilbert function H_L(I) of the leading ideal L(I) coincide. @end ifinfo @end table @c --------------------------------------------------------------------------- @node Hilbert function, Syzygies and resolutions, Standard bases, Mathematical background @section Hilbert function @cindex Hilbert function @cindex Hilbert series @tex Let M $=\bigoplus_i M_i$ be a graded module over $K[x_1,..,x_n]$ with respect to weights $(w_1,..w_n)$. The {\bf Hilbert function} of $M$, $H_M$, is defined (on the integers) by $$H_M(k) :=dim_K M_k.$$ The {\bf Hilbert-Poincare series} of $M$ is the power series $$\hbox{HP}_M(t) :=\sum_{i=-\infty}^\infty H_M(i)t^i=\sum_{i=-\infty}^\infty dim_K M_i \cdot t^i.$$ It turns out that $\hbox{HP}_M(t)$ can be written in two useful ways for weights $(1,..,1)$: $$\hbox{HP}_M(t)={Q(t)\over (1-t)^n}={P(t)\over (1-t)^{dim(M)}}$$ where $Q(t)$ and $P(t)$ are polynomials in ${\bf Z}[t]$. $Q(t)$ is called the {\bf first Hilbert series}, and $P(t)$ the {\bf second Hilbert series}. If \hbox{$P(t)=\sum_{k=0}^N a_k t^k$}, and \hbox{$d = dim(M)$}, then \hbox{$H_M(s)=\sum_{k=0}^N a_k$ ${d+s-k-1}\choose{d-1}$} (the {\bf Hilbert polynomial}) for $s \ge N$. @end tex @ifinfo Let M =(+) M_i be a graded module over K[x_1,...,x_n] with respect to weights (w_1,..w_n). The Hilbert function of M H_M is defined by @display H_M(k)=dim_K M_k. @end display The Hilbert-Poincare series of M is the power series @display HP_M(t)=sum_i dim_K (M_i)*t^i. @end display It turns out that HP_M(t) can be written in two useful ways for weights $(1,..,1)$: @display H_M(t)=Q(t)/(1-t)^n=P(t)/(1-t)^dim(M). @end display where Q(t) and P(t) are polynomials in Z[t]. Q(t) is called the first Hilbert series, and P(t) the second Hilbert series. If P(t)=sum_(k=0)^N a_k t^k, and d=dim(M), then @display H_M(s)=sum_(k=0)^N a_k binomial(d+s-k-1,d-1) (the Hilbert polynomial) @end display for s >= N. @end ifinfo @* @* @tex Generalizing these to quasihomogeneous modules we get $$\hbox{HP}_M(t)={Q(t)\over {\Pi_{i=1}^n(1-t^{w_i})}}$$ where $Q(t)$ is a polynomial in ${\bf Z}[t]$. $Q(t)$ is called the {\bf first (weighted) Hilbert series} of M. @end tex @ifinfo Generalizing these to quasihomogeneous modules we get @display H_M(t)=Q(t)/Prod((1-t)^(w_i)). @end display where Q(t) is a polynomial in Z[t]. Q(t) is called the first (weighted) Hilbert series of M. @end ifinfo @c --------------------------------------------------------------------------- @node Syzygies and resolutions, Characteristic sets, Hilbert function, Mathematical background @section Syzygies and resolutions @cindex Syzygies and resolutions @subheading Syzygies @tex Let $R$ be a quotient of $\hbox{Loc}_< K[\underline{x}]$ and let \hbox{$I=(g_1, ..., g_s)$} be a submodule of $R^r$. Then the {\bf module of syzygies} (or {\bf 1st syzygy module}, {\bf module of relations}) of $I$, syz($I$), is defined to be the kernel of the map \hbox{$R^s \rightarrow R^r,\; \sum_{i=1}^s w_ie_i \mapsto \sum_{i=1}^s w_ig_i$.} @end tex @ifinfo Let R be a quotient of Loc K[x] and let I=(g_1, ..., g_s) be a submodule of R^r. Then the @strong{module of syzygies} (or @strong{1st syzygy module}, @strong{module of relations}) of I, syz(I), is defined to be the kernel of the map @display R^s --> R^r, w_1*e_1 + ... + w_s*e_s -> w_1*g_1 + ... + w_s*g_s. @end display @end ifinfo The @strong{k-th syzygy module} is defined inductively to be the module of syzygies of the @tex $(k-1)$-st @end tex @ifinfo (k-1)-st @end ifinfo syzygy module. @tex Note, that the syzygy modules of $I$ depend on a choice of generators $g_1, ..., g_s$. 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, ..., g_s. But one can show that they depend on I uniquely up to direct summands. @end ifinfo @table @code @item @strong{Example:} @smallexample @c example ring R= 0,(u,v,x,y,z),dp; ideal i=ux, vx, uy, vy; print(syz(i)); @c example @end smallexample @end table @subheading Free resolutions @tex Let $I=(g_1,...,g_s)\subseteq R^r$ and $M= R^r/I$. A {\bf free resolution of $M$} is a long exact sequence $$...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F_1 \buildrel{A_1}\over{\longrightarrow} F_0\longrightarrow M\longrightarrow 0,$$ @end tex @ifinfo Let I=(g_1,...,g_s) in R^r and M=R^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 $A_1$ @end tex @ifinfo A_1 @end ifinfo generate @math{I}. Note, that resolutions need not to be finite (i.e., of finite length). The Hilbert Syzygy Theorem states, that for @tex $R=\hbox{Loc}_< K[\underline{x}]$ @end tex @ifinfo R=Loc K[x] @end ifinfo there exists a ("minimal") resolution of length not exceeding the number of variables. @table @code @item @strong{Example:} @smallexample @c example ring R= 0,(u,v,x,y,z),dp; ideal I = ux, vx, uy, vy; 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[3])); @c example @end smallexample @end table @subheading Betti numbers and regularity @cindex Betti number @cindex regularity @tex Let $R$ be a graded ring (e.g., $R = \hbox{Loc}_< K[\underline{x}]$) and let $I \subset R^r$ be a graded submodule. Let $$ R^r = \bigoplus_a R\cdot e_{a,0} \buildrel A_1 \over \longleftarrow \bigoplus_a R\cdot e_{a,1} \longleftarrow \ldots \longleftarrow \bigoplus_a R\cdot e_{a,n} \longleftarrow 0 $$ be a minimal free resolution of $R^n/I$ considered with homogeneous maps of degree 0. Then the {\bf graded Betti number} $b_{i,j}$ of $R^r/I$ is the minimal number of generators $e_{a,j}$ in degree $i+j$ of the $j$-th syzygy module of $R^r/I$ (i.e., the $(j-1)$-st syzygy module of $I$). Note, that by definition the $0$-th syzygy module of $R^r/I$ is $R^r$ and the 1st syzygy module of $R^r/I$ is $I$. @end tex @ifinfo Let R be a graded ring (e.g., R = K[x]) and let I in R^r be a graded submodule. Let @display R^r = (+) K[x]e(a,0) <--- (+) K[x]e(a,1) <--- @dots{} <--- (+) K[x]e(a,n) <--- 0 @end display be a minimal free resolution of R^n/I considered with homogeneous maps of degree 0. Then the @strong{graded Betti number} b_i,j of R^r/I is the minimal number of generators e_a,j in degree i+j of the j-th syzygy module of R^r/I (i.e., the (j-1)-st syzygy module of I). Note, that by definition the 0th syzygy module of R^r/I is R^r and the 1st syzygy module of R^r/I is I. @end ifinfo The @strong{regularity} of @math{I} is the smallest integer @math{s} such that @tex $$ \hbox{deg}(e_{a,j}) \le s+j-1 \quad \hbox{for all $j$.} $$ @end tex @ifinfo @display deg(e(a,j)) <= s+j-1 for all j. @end display @end ifinfo @table @code @item @strong{Example:} @smallexample @c example ring R= 0,(u,v,x,y,z),dp; ideal I = ux, vx, uy, vy; resolution resI = mres(I,0); resI; // the betti number: print(betti(resI), "betti"); // the regularity: regularity(resI); @c example @end smallexample @end table @c --------------------------------------------------------------------------- @node Characteristic sets, Gauss-Manin connection, Syzygies and resolutions, Mathematical background @section Characteristic sets @cindex Characteristic sets @tex Let $<$ be the lexicographical ordering on $R=K[x_1,...,x_n]$ with $x_1 < ... < x_n$. For $f \in R$ let lvar($f$) (the leading variable of $f$) be the largest variable in $f$, i.e., if $f=a_s(x_1,...,x_{k-1})x_k^s+...+a_0(x_1,...,x_{k-1})$ for some $k \leq n$ then lvar$(f)=x_k$. Moreover, let \hbox{ini}$(f):=a_s(x_1,...,x_{k-1})$. The pseudoremainder $r=\hbox{prem}(g,f)$ of $g$ with respect to $f$ is defined by the equality $\hbox{ini}(f)^a\cdot g = qf+r$ with $\hbox{deg}_{lvar(f)}(r)<\hbox{deg}_{lvar(f)}(f)$ and $a$ minimal. A set $T=\{f_1,...,f_r\} \subset R$ is called triangular if $\hbox{lvar}(f_1)<...<\hbox{lvar}(f_r)$. Moreover, let $ U \subset T $, then $(T,U)$ is called a triangular system, if $T$ is a triangular set such that $\hbox{ini}(T)$ does not vanish on $V(T) \setminus V(U) (=:V(T\setminus U))$. $T$ is called irreducible if for every $i$ there are no $d_i$,$f_i'$,$f_i''$ such that $$ \hbox{lvar}(d_i)<\hbox{lvar}(f_i) = \hbox{lvar}(f_i')=\hbox{lvar}(f_i''),$$ $$ 0 \not\in \hbox{prem}(\{ d_i, \hbox{ini}(f_i'), \hbox{ini}(f_i'')\},\{ f_1,...,f_{i-1}\}),$$ $$\hbox{prem}(d_if_i-f_i'f_i'',\{f_1,...,f_{i-1}\})=0.$$ Furthermore, $(T,U)$ is called irreducible if $T$ is irreducible. The main result on triangular sets is the following: let $G=\{g_1,...,g_s\} \subset R$ then there are irreducible triangular sets $T_1,...,T_l$ such that $V(G)=\bigcup_{i=1}^{l}(V(T_i\setminus I_i))$ where $I_i=\{\hbox{ini}(f) \mid f \in T_i \}$. Such a set $\{T_1,...,T_l\}$ is called an {\bf irreducible characteristic series} of the ideal $(G)$. @end tex @ifinfo Let > be the lexicographical ordering on R=K[x_1,...,x_n] with x_1<...), i.e., if f=a_s(x_1,...,x_(k-1))x_k^s+...+a_0(x_1,...,x_(k-1)) for some k<=n then lvar(f)=x_k. Moreover, let ini(f):=a_s(x_1,...,x_(k-1)). The pseudoremainder r=prem(g,f) of g with respect to f is defined by ini(f)^a*g=q*f+r with the property deg_(lvar(f))(r)(C,0) be a complex isolated hypersurface singularity given by a polynomial with algebraic coefficients which we also denote by f. Let O=C[x_0,...,x_n]_(x_0,...,x_n) be the local ring at the origin and J_f the Jacobian ideal of f. A @strong{Milnor representative} of f defines a differentiable fibre bundle over the punctured disc with fibres of homotopy type of mu n-spheres. The n-th cohomology bundle is a flat vector bundle of dimension n and carries a natural flat connection with covariant derivative d_t. The @strong{monodromy operator} is the action of a positively oriented generator of the fundamental group of the puctured disc on the Milnor fibre. Sections in the cohomology bundle of @strong{moderate growth} at 0 form a regular D=C@{t@}[d_t]-module G, the @strong{Gauss-Manin connection}. By integrating along flat multivalued families of cycles, one can consider fibrewise global holomorphic differential forms as elements of G. This factors through an inclusion of the @strong{Brieskorn lattice} H'':=Omega^(n+1)_(C^(n+1),0)/df*dOmega^(n-1)_(C^(n+1),0) in G. The D-module structure defines the @strong{V-filtration} V on G by V^a:=sum_(b>=a)C@{t@}ker(t*d_t-b)^(n+1). The Brieskorn lattice defines the @strong{Hodge filtration} F on G by F_k=d_t^kH'' which comes from the @strong{mixed Hodge structure} on the Milnor fibre. Note that F_(-1)=H'. The induced V-filtration on the Brieskorn lattice determines the @strong{singularity spectrum} Sp by Sp(a):=dim_CGr_V^a Gr^F_0G. The spectrum consists of mu rational numbers a_1,...,a_mu such that exp(2*pi*i*a_1),...,exp(2*pi*i*a_mu) are the eigenvalues of the monodromy. These @strong{spectral numbers} lie in the open interval (-1,n), symmetric about the midpoint (n-1)/2. The spectrum is constant under mu-constant deformations and has the following semicontinuity property: The number of spectral numbers in an interval (a,a+1] of all singularities of a small deformation of f is greater or equal to that of f in this interval. For semiquasihomogeneous singularities, this also holds for intervals of the form (a,a+1). Two given isolated singularities f and g determine two spectra and from these spectra we get an integer. This integer is the maximal positive integer k such that the semicontinuity holds for the spectrum of f and k times the spectrum of g. These numbers give bounds for the maximal number of isolated singularties of a specific type on a hypersurface X in P^n of degree d: such a hypersurface has a smooth hyperplane section, and the complement is a small deformation of a cone over this hyperplane section. The cone itself being a mu-constant deformation of x_0^d+...+x_n^d=0, the singularities are bounded by the spectrum of x_0^d+...+x_n^d. Using the library @code{gmssing.lib} one can compute the @strong{monodromy}, the V-fitration on H''/H', and the spectrum. @end ifinfo Let us consider as an example @math{f=x^5+x^2y^2+y^5}. First, we compute a matrix @math{M} such that @tex $\exp(2\pi iM)$ @end tex @ifinfo exp(-2*pi*i*M) @end ifinfo is a monodromy matrix of @math{f} and the Jordan normal form of @math{M}: @smallexample // not computed !! LIB "gmssing.lib"; ring R=0,(x,y),ds; poly f=x5+x2y2+y5; matrix M=monodromy(f); print(jordanform(M)); @end smallexample Now, we compute the V-fitration on @math{H''/H'} and the spectrum: @smallexample // not computed !! LIB "gmssing.lib"; ring R=0,(x,y),ds; poly f=x5+x2y2+y5; list l=vfiltration(f); print(l[1]); print(l[2]); print(l[3]); print(l[4]); @end smallexample Here @code{l[1]} contains the spectral numbers, @code{l[2]} the corresponding multiplicities, @code{l[3]} a @math{C}-basis of the V-filtration on @math{H''/H'} in terms of the monomial basis of @tex $O/J_f\cong H''/H'$ @end tex @ifinfo O/J_f~=H''/H' @end ifinfo in @code{l[4]}. @tex If the principal part of $f$ is $C$-nondegenerate, one can compute the spectrum using the library {\tt spectrum.lib}. In this case, the V-filtration on $H''$ coincides with the Newton-filtration on $H''$ which allows to compute the spectrum more efficiently. @end tex @ifinfo If the principal part of f is C-nondegenerate, one can compute the spectrum using the library @code{spectrum.lib}. In this case, the V-filtration on H'' coincides with the Newton-filtration on H'' which allows to compute the spectrum more efficiently. @end ifinfo Let us calculate one specific example, the maximal number of triple points of type @tex $\tilde{E}_6$ on a surface $X\subset{P}^3$ @end tex @ifinfo E~_6 on a surface X in P^3 @end ifinfo of degree seven. This calculation can be done over the rationals. So choose a local ordering on @math{Q[x,y,z]}. Here we take the negative degree lexicographical ordering which is denoted @code{ds} in @sc{Singular}: @smallexample @c example ring r=0,(x,y,z),ds; LIB "spectrum.lib"; poly f=x^7+y^7+z^7; list s1=spectrumnd( f ); s1; @c example @end smallexample The command @code{spectrumnd(f)} computes the spectrum of @math{f} and returns a list with six entries: The Milnor number @tex $\mu(f)$, the geometric genus $p_g(f)$ @end tex @ifinfo mu(f), the geometric genus p_g(f) @end ifinfo and the number of different spectrum numbers. The other three entries are of type @code{intvec}. They contain the numerators, denominators and multiplicities of the spectrum numbers. So @tex $x^7+y^7+z^7=0$ @end tex @ifinfo x^7+y^7+z^7=0 @end ifinfo has Milnor number 216 and geometrical genus 35. Its spectrum consists of the 16 different rationals @* @tex ${3 \over 7}, {4 \over 7}, {5 \over 7}, {6 \over 7}, {1 \over 1}, {8 \over 7}, {9 \over 7}, {10 \over 7}, {11 \over 7}, {12 \over 7}, {13 \over 7}, {2 \over 1}, {15 \over 7}, {16 \over 7}, {17 \over 7}, {18 \over 7}$ @end tex @ifinfo 3/7, 4/7, 5/7, 6/7, 1, 8/7, 9/7, 10/7, 11/7, 12/7, 13/7, 2, 15/7, 16/7, 17/7, 18/7 @end ifinfo @*appearing with multiplicities @*1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1. @tex The singularities of type $\tilde{E}_6$ form a $\mu$-constant one parameter family given by $x^3+y^3+z^3+\lambda xyz=0,\quad \lambda^3\neq-27$. @end tex @ifinfo The singularities of type E~_6 form a mu-constant one parameter family given by x^3+y^3+z^3+lambda xyz=0, lambda^3 <> -27. @end ifinfo Therefore they have all the same spectrum, which we compute for @tex $x^3+y^3+z^3$. @end tex @ifinfo @math{x^3+y^3+z^3}. @end ifinfo @smallexample poly g=x^3+y^3+z^3; list s2=spectrumnd(g); s2; @expansion{} [1]: @expansion{} 8 @expansion{} [2]: @expansion{} 1 @expansion{} [3]: @expansion{} 4 @expansion{} [4]: @expansion{} 1,4,5,2 @expansion{} [5]: @expansion{} 1,3,3,1 @expansion{} [6]: @expansion{} 1,3,3,1 @end smallexample Evaluating semicontinuity is very easy: @smallexample semicont(s1,s2); @expansion{} 18 @end smallexample This tells us that there are at most 18 singularities of type @tex $\tilde{E}_6$ on a septic in $P^3$. But $x^7+y^7+z^7$ @end tex @ifinfo E~_6 on a septic in P^3. But x^7+y^7+z^7 @end ifinfo is semiquasihomogeneous (sqh), so we can also apply the stronger form of semicontinuity: @smallexample semicontsqh(s1,s2); @expansion{} 17 @end smallexample So in fact a septic has at most 17 triple points of type @tex $\tilde{E}_6$. @end tex @ifinfo E~_6. @end ifinfo Note that @code{spectrumnd(f)} works only if @math{f} has nondegenerate principal part. In fact @code{spectrumnd} will detect a degenerate principal part in many cases and print out an error message. However if it is known in advance that @math{f} has nondegenerate principal part, then the spectrum may be computed much faster using @code{spectrumnd(f,1)}. @c --------------------------------------------------------------------------- @ifclear withplural @node Toric ideals and integer programming, References, Gauss-Manin connection, Mathematical background @end ifclear @ifset withplural @node Toric ideals and integer programming, Non-commutative algebra , Gauss-Manin connection, Mathematical background @end ifset @section Toric ideals and integer programming @cindex Toric ideals and integer programming @include ti_ip.tex @ifset withplural @c --------------------------------------------------------------------------- @node Non-commutative algebra, References, Toric ideals and integer programming, Mathematical background @section Non-commutative algebra @cindex Non-commutative algebra @xref{Mathematical background (plural)}. @end ifset @c --------------------------------------------------------------------------- @ifclear withplural @node References, , Toric ideals and integer programming, Mathematical background @end ifclear @ifset withplural @node References, ,Non-commutative algebra , Mathematical background @end ifset @section References @cindex References 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.org/}, @code{http://www-sop.inria.fr/galaad/}, @code{http://www.bath.ac.uk/~masjpf/CAIN.html},... and the following list of books. @ifset withplural For the references for noncommutative algebras and algorithms, see @ref{References (plural)}. @end ifset @subheading Text books on computational algebraic geometry @itemize @bullet @item Adams, W.; Loustaunau, P.: An Introduction to Gr@"obner Bases. Providence, RI, AMS, 1996 @item Becker, T.; Weisspfenning, V.: Gr@"obner Bases - A Computational Approach to Commutative Algebra. Springer, 1993 @item Cohen, H.: A Course in Computational Algebraic Number Theory, Springer, 1995 @item Cox, D.; Little, J.; O'Shea, D.: Ideals, Varieties and Algorithms. Springer, 1996 @item Cox, D.; Little, J.; O'Shea, D.: Using Algebraic Geometry. Springer, 1998 @item Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, 1995 @item Greuel, G.-M.; Pfister, G.: A Singular Introduction to Commutative Algebra. Springer, 2002 @item Mishra, B.: Algorithmic Algebra, Texts and Monographs in Computer Science. Springer, 1993 @item Sturmfels, B.: Algorithms in Invariant Theory. Springer 1993 @item Vasconcelos, W.: Computational Methods in Commutative Algebra and Algebraic Geometry. Springer, 1998 @end itemize @subheading Descriptions of algorithms @itemize @bullet @item Bareiss, E.: Sylvester's identity and multistep integer-preserving Gaussian elimination. Math. Comp. 22 (1968), 565-578 @item Campillo, A.: Algebroid curves in positive characteristic. SLN 813, 1980 @item Chou, S.: Mechanical Geometry Theorem Proving. D.Reidel Publishing Company, 1988 @item Decker, W.; Greuel, G.-M.; Pfister, G.: Primary decomposition: algorithms and comparisons. Preprint, Univ. Kaiserslautern, 1998. To appear in: Greuel, G.-M.; Matzat, B. H.; Hiss, G. (Eds.), Algorithmic Algebra and Number Theory. Springer Verlag, Heidelberg, 1998 @item Decker, W.; Greuel, G.-M.; de Jong, T.; Pfister, G.: The normalisation: a new algorithm, implementation and comparisons. Preprint, Univ. Kaiserslautern, 1998 @item Decker, W.; Heydtmann, A.; Schreyer, F. O.: Generating a Noetherian Normalization of the Invariant Ring of a Finite Group, 1997, to appear in Journal of Symbolic Computation @item @tex Faug\`ere, @end tex @ifinfo Faugere, @end ifinfo J. C.; Gianni, P.; Lazard, D.; Mora, T.: Efficient computation of zero-dimensional Gr@"obner bases by change of ordering. Journal of Symbolic Computation, 1989 @item Gr@"abe, H.-G.: On factorized Gr@"obner bases, Univ. Leipzig, Inst. f@"ur Informatik, 1994 @item Grassmann, H.; Greuel, G.-M.; Martin, B.; Neumann, W.; Pfister, G.; Pohl, W.; Sch@"onemann, H.; Siebert, T.: On an implementation of standard bases and syzygies in @sc{Singular}. Proceedings of the Workshop Computational Methods in Lie theory in AAECC (1995) @item Greuel, G.-M.; Pfister, G.: Advances and improvements in the theory of standard bases and syzygies. Arch. d. Math. 63(1995) @item Kemper; Generating Invariant Rings of Finite Groups over Arbitrary Fields. 1996, to appear in Journal of Symbolic Computation @item Kemper and Steel: Some Algorithms in Invariant Theory of Finite Groups. 1997 @item Lee, H.R.; Saunders, B.D.: Fraction Free Gaussian Elimination for Sparse Matrices. Journal of Symbolic Computation (1995) 19, 393-402 @item Sch@"onemann, H.: Algorithms in @sc{Singular}, Reports on Computer Algebra 2(1996), Kaiserslautern @item Siebert, T.: On strategies and implementations for computations of free resolutions. Reports on Computer Algebra 8(1996), Kaiserslautern @item Wang, D.: Characteristic Sets and Zero Structure of Polynomial Sets. Lecture Notes, RISC Linz, 1989 @end itemize