H. Schönemann; Humboldt-Universität $3 \times 3$ $0$ $Q$ $a$ $Z/32003[x,y,z]$ $x^3$ $K[x,y,z]/\hbox{\rm jacob}(f)$. $12-4=8$. $1 \times 1$-minors $1 \times 1$-minors, $MD$ $r^3/MD$. $M = r^3/U$, $M$. $\hbox{ann}(M) = \{a \mid aM = 0 \}$ $\{ a \mid ar^3 \in U \}$. $U \colon r^3 $. $n$ $n=0$ $R^5$ $(z^3,0,-y+4z,x+2z,0)$ and $(-xyz-y^2z-4xz^2+16z^3,-y^2,48z,48z,x+y-z)$. finite fields $Z/p$, $p$ a prime $\le 2147483629$, finite fields $\hbox{GF}(p^n)$ with $p^n$ elements, $p$ a prime, $p^n \le 2^{15}$, $Z/p$ $Q[a,b,c,d]$ $Z/7[x,y,z]$ $Z/7[x_1,\ldots,x_6]$ $x_1,x_2,x_3$ $x_4,x_5,x_6$: $(Q[a,b,c])[x,y,z]$ $(x,y,z)$ $Q[x,y,z]$ $x$ $y$ $z$ $K[x,y,z]$ $K=Z/7(a,b,c)$ $Z/7$ $b$ $c$ $K=Z/7[a]$ $a.$ $K$ $\mu_a=a^2+a+3$, $R[x,y,z]$ $R$ $R(j)[x,y,z]$ $j$ $R(i)[x,y,z]$ $i$ $\hbox{GF}(p^n)$ with $p^n$ elements, where $p^n$ has to be smaller or equal $2^{15}$. $\hbox{GF}(p^n)$ $K[x_1,\ldots,x_n]$, $\hbox{Loc}_{(x)}K[x_1,\ldots,x_n])$. $1 < x$ $ m \times n $      $\alpha$ $ i_{1,1} $ $i+i+i$ $\alpha, \beta$ $i_1$ $ i_{1,1} $ $(h1+h2)/h1 \cong h2/(h1 \cap h2)$ $Q \rightarrow Q(a, \ldots)$ $Q \rightarrow R$ $Q \rightarrow C$ $Z/p \rightarrow (Z/p)(a, \ldots)$ $Z/p \rightarrow GF(p^n)$ $Z/p \rightarrow R$ $R \rightarrow C$ $Z/p \rightarrow Q,
\quad
[i]_p \mapsto i \in [-p/2, \, p/2]
\subseteq Z$ $Z/p \rightarrow Z/p^\prime,
\quad
[i]_p \mapsto i \in [-p/2, \, p/2] \subseteq Z, \;
i \mapsto [i]_{p^\prime} \in Z/p^\prime$ $C \rightarrow R, \quad$ the real part $Q \rightarrow Z/p$ $Q \rightarrow (Z/p)(a, \ldots)$ $M$ $R^n$, $v_1, \ldots, v_k$, then $v_1, \ldots, v_k$ $R^n/M$ n$\times$k R$^n$/M, $v_1, \ldots, v_k$ $(h1+h2)/h1=h2/(h1 \cap h2)$ $k[X,Y]$ $k_1[X]$ $k_2[Y]$ $k_1$ $k_2$ $C$ $R^n/M$, if $R$ denotes the basering and $M$ a homogeneous submodule of $R^n$ and the argument represents a resolution of $R^n/M$. The entry d of the intmat at place (i,j) is the minimal number of generators in degree i+j of the j-th syzygy module (= module of relations) of $R^n/M$ (the 0th (resp. 1st) syzygy module of $R^n/M$ is $R^n$ (resp. $M$)).

\begin{displaymath}
0 \longleftarrow r/j \longleftarrow r(1)
\buildrel{T[1]}\ove...
...ldrel{T[3]}\over{\longleftarrow} r(5)
\longleftarrow 0 \quad .
\end{displaymath}

The third argument is used to return the matrix T of coefficients such that matrix(J) = T*M. $M=(m_{ij})$

\begin{displaymath}J_j = z^0 \cdot m_{1j} + z^1 \cdot m_{2j} + ... + z^{d-1} \cdot m_{dj},\end{displaymath}

while for a module J the i-th component of the j-th generator is equal to the entry [i,j] of matrix(J), and we get

\begin{displaymath}J_{i,j} = z^0 \cdot m_{(i-1)d+1,j} + z^1 \cdot m_{(i-1)d+2,j} + ... +
z^{d-1} \cdot m_{id,j}.\end{displaymath}


\begin{displaymath}{\rm contract}(x^A , x^B) := \cases{ x^{(B-A)}, &if $B\ge A$
componentwise\cr 0,&otherwise.\cr}\end{displaymath}

$x$. $x_i$ $x_i>1$ then $x_i$ $f_1,\dots,f_k$ of I, let $f'_i$ be obtained from $f_i$ by deleting the terms divisible by $x_i\cdot m$ for all i with $x_i<1$. Then $f'_1,\dots,f'_k$ generate I.

\begin{displaymath}...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F...
...}\over{\longrightarrow} R\longrightarrow R/I
\longrightarrow 0.\end{displaymath}

$M_i={\tt module} (A_i)$, i=1..k. ${\tt L[i]}=M_i$ $I \cap K[U]=(0)$, input: $f_1,\dots,f_n$ output: $g_1,\dots,g_s$ with $s \leq n$ and the properties $(f_1,\dots,f_n) = (g_1,\dots,g_s)$ $L(g_i)\neq L(g_j)$ for all $i\neq j$ $L(g_i)$ $\{g_1,\dots,g_{i-1},g_{i+1},\dots,g_s\}$ $L(g_i) \vert L(g_j)$ for any $i\neq j$, $ecart(g_i) > ecart(g_j)$ Here, $L(g)$ denotes the leading term of $g$ and $ecart(g):=deg(g)-deg(L(g))$. $M_i={\tt module} (A_i)$, i=1..k. represents $h_1/(h_1 \cap h_2) \cong (h_1+h_2)/h_2$ $h_1$ and $h_2$ $R^l$ $H_1$, resp. $H_2$, be the matrices of size $l \times k$, resp. $l \times m$, having the generators of $h_1$, resp. $h_2$, $h_1/(h_1 \cap h_2) \cong R^k / ker(\overline{H_1})$ $\overline{H_1}: R^k \rightarrow R^l/Im(H_2)=R^l/h_2$ is the induced map. $coker(A)=F_0/M$

\begin{displaymath}...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F...
...er{\longrightarrow} F_0\longrightarrow F_0/M
\longrightarrow 0,\end{displaymath}

$A_1$ $M_i={\tt module} (A_i)$, i=1...k. ${\tt L[i]}\neq 0$ for $i \le p$, $A_1$=matrix(M), $coker(A_1)=F_0/M$

\begin{displaymath}...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F...
...er{\longrightarrow} F_0\longrightarrow F_0/M
\longrightarrow 0,\end{displaymath}

( ${\tt L[i]}=M_i$ $d$ in row $i$ and column $j$ $i+j$ of the $j$-th $R^n/M$ (the 0th and 1st syzygy module of $R^n/M$ is $R^n$ and $M$, resp.). ${\tt R}^n$. $\{a \in R \mid aJ \subset I\}$, $\{b \in R^n \mid bJ \subset M\}$. Let $0 \rightarrow\ \bigoplus_a K[x]e_{a,n}\ \rightarrow\ \dots
\rightarrow\ \bigoplus_a K[x]e_{a,0}\ \rightarrow\
I\ \rightarrow\ 0$ be a minimal resolution of I considered with homogeneous maps of degree 0. The regularity is the smallest number $s$ with the property deg( $e_{a,i})
\leq s+i$ for all $i$. computes the permutation v which orders the ideal, resp. module, I by its initial terms, starting with the smallest, that is, I(v[i]) < I(v[i+1]) for all i. $A_1={\tt matrix}(M)$. $coker(A_1)=F_0/M$

\begin{displaymath}...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F...
...ver{\longrightarrow} F_0\longrightarrow F_0/M\longrightarrow 0.\end{displaymath}

vandermonde(p,v,d) computes the (unique) polynomial of degree @coded with prescribed values v[1],...,v[N] at the points p$^0,\dots,$ p$^{N-1}$, N=(d+1)$^n$, $n$ the number of ring variables.

The returned polynomial is $\sum
c_{\alpha_1\ldots\alpha_n}\cdot x_1^{\alpha_1} \cdot \dots \cdot
x_n^{\alpha_n}$, where the coefficients $c_{\alpha_1\ldots\alpha_n}$ are the solution of the (transposed) Vandermonde system of linear equations

\begin{displaymath}\sum_{\alpha_1+\ldots+\alpha_n\leq d} c_{\alpha_1\ldots\alpha...
...\tt p}_n^{(k-1)\alpha_n} =
{\tt v}[k], \quad k=1,\dots,{\tt N}.\end{displaymath}

the ground field has to be the field of rational numbers. Moreover, ncols(p)==$n$, the number of variables in the basering, and all the given generators have to be numbers different from 0,1 or -1. Finally, ncols(v)==(d+1)$^n$, and all given generators have to be numbers. $K[[x_1,\ldots,x_n]]$

\begin{displaymath}
\hbox{milnor}(f) = \hbox{dim}_K(K[[x_1,\ldots,x_n]]/\hbox{jacob}(f)),
\end{displaymath}

respectively

\begin{displaymath}
\hbox{tjurina}(f) = \hbox{dim}_K(K[[x_1,\ldots,x_n]]/((f)+\hbox{jacob}(f)))
\end{displaymath}

where $\hbox{Loc}_{(x)}K[x_1,\ldots,x_n]$, $K[x_1,\ldots,x_n]$ $(x_1,\ldots,x_n)$. $\hbox{Loc}_{(x)}K[x_1,\ldots,x_n])$ $\hbox{dim}_K(\hbox{Loc}_{(x,y,z)}K[x,y,z]/\hbox{jacob}(f))$ $\hbox{dim}_K(K[x,y,z]/\hbox{jacob}(f))$ $\hbox{dim}_K(\hbox{Loc}_{(x,y,z)}K[x,y,z]/(\hbox{jacob}(f)+(f)))$ $\hbox{dim}_K(K[x,y,z]/(\hbox{jacob}(f)+(f)))$, $j+(f)$ $f=0$ $m$ $i, j$

\begin{displaymath}
\hbox{sat}(i,j)=\{x\in R\;\vert\; \exists\;n\hbox{ s.t. }
x\cdot(j^n)\subseteq i\}
= \bigcup_{n=1}^\infty i:j^n\end{displaymath}

$sat(j+(f),m)$ $dim_{Q(t)}Q(t)[x,y]/j$. $a \in Q$ $dim_Q Q[x,y]/j_0$, $j_0=j\vert _{t=a}$. $T^1$ $T^2$ $h_1,\ldots,h_r$ $I \subset R$ is the ideal generated by $f_1,...,f_s$, then any infinitesimal deformation of $R/I$ over $K[\varepsilon]/(\varepsilon^2)$ is given by $f+\varepsilon g$, where $f=(f_1,...,f_s)$, $g$ a $K$-linear combination of the $h_i$. $d$ $p^k$ $f=(f_1,\ldots,f_n):k^r\rightarrow k^n$

\begin{displaymath}
\displaylines{
j=J \cap k[x_1,\ldots,x_n], \;\quad\hbox{\rm ...
...n(t_1,\ldots,t_r))\subseteq
k[t_1,\ldots,t_r,x_1,\ldots,x_n]
}
\end{displaymath}

$t_1,\ldots,t_r$. $f:(k^r,0)\rightarrow(k^n,0)$, $k^r\times(k^n,0)$,

\begin{displaymath}\hbox{pr}:k^r\times(k^n,0)\rightarrow(k^n,0)\end{displaymath}

can be computed. $\hbox{T}[7]^\prime$ $P^4$. $K^3$ $K^3$. $P^4$ $\hbox{\rm ker} / \hbox{\rm Im}$ $(n-1)$ $A/(A \cap B)$ $\hbox{Ext}^1(M,M)$, resp. $\hbox{Ext}^2(M,M)$, $\hbox{Ext}^1$ $\hbox{Ext}^1$, $\hbox{Ext}^2$ $\hbox{Ext}^k(R/J,R)$ $J$ ( $=\hbox{Ext}^1(K,K)$) ( $=\hbox{Ext}^2(K,K)$) $K=R/m$ $R=Loc_m K[x,y]/(x^2-y^3)$, $m=(x,y)$. $\hbox{Ext}^1(m,m)$ $\hbox{Ext}^2(K,K)$. $\hbox{Ext}^k(R/i,R)$ $f\in k[x_1,\ldots,x_n,t]$ $t$ $V(\partial f/\partial x_1,\ldots,\partial f/\partial x_n) \setminus V(f)$ $V(\partial f/\partial x_1,\ldots,\partial f/\partial x_n)$ (in $P^n$), (in $k^n$) (in $(k^n,0)$), D$_k$(R) $(m+1)$ $(n \times n)$-matrix. $\hbox{depth}(\hbox{D}_k(R))\geq m(m+1)/2 + m-1$ $\sum x_i \partial /\partial x_{i+1}$. $f$ $R/I$ $A$ $B$ $m\times r$ and $m\times s$

\begin{displaymath}
R^r \buildrel{A}\over{\longrightarrow}
R^m \buildrel{B}\over{\longleftarrow} R^s\;.
\end{displaymath}

$R^r \buildrel{A}\over{\longrightarrow}
R^m\longrightarrow
R^m/\hbox{Im}(B) \;.$

\begin{displaymath}
\hbox{\tt modulo}(A,B)=\hbox{ker}(R^r
\buildrel{A}\over{\longrightarrow}R^m/\hbox{Im}(B)) \; .
\end{displaymath}

$g$, $f_1$, ..., $f_r\in K[x_1,\ldots,x_n]$. $f_1$, ..., $f_r$ $I=\langle Y_1-f_1,\ldots,Y_r-f_r \rangle \subseteq
K[x_1,\ldots,x_n,Y_1,\ldots,Y_r]$. $I \cap K[Y_1,\ldots,Y_r]$ $f_1$, ..., $f_r$. $g \in K [f_1,\ldots,f_r]$. $g \in K [f_1,\ldots,f_r]$ $g$ $I$ $X=(x_1,\ldots,x_n)$ and $Y=(Y_1,\ldots,Y_r)$ with $X>Y$ $K[Y]$ $[f_1,...,f_n]$ or $f_1*gen(1)+...+f_n*gen(n)$, where $gen(i)$ $gen(i)$ $[f_1,...,f_n]$ $v=[f_1,...,f_n]$ nrows($v$) $r$ $f_r \not= 0$. $v$ $f_i*gen(i)$ $f_i=0$ $v_1,...,v_k$ nrows($M$) nrows($v_i$). $R$. $N$, $N = R^n/M$ where $n$ $N$ and $M \subseteq R^n$ $R^n$ $N = R^n/M$. $M$, $N = R^n/M$ instead of $M$. $N$ as $N = R^n/$std($M$)). dim$(R^n/M)$, resp.@: dim$_k(R^n/M)$ $N$. $N = R^n/M$ $\hbox{Loc}_{(x)}K[x_1,\ldots,x_n]$. $\hbox{Wp}(w_1, \ldots, w_n)$ $\hbox{Ws}(w_1, \ldots, w_n)$) $w_1$, $\ldots$, $w_n$ of $x_1$, $\ldots$, $x_n$. A monomial ordering (term ordering) on $K[x_1,\ldots,x_n]$ is a total ordering $<$ on the set of monomials (power products) $\{x^\alpha \mid \alpha \in \bf {N}^n\}$ which is compatible with the natural semigroup structure, i.e., $x^\alpha < x^\beta$ implies $x^\gamma
x^\alpha < x^\gamma x^\beta$ for any $\gamma \in \bf {N}^n$. We do not require $<$ to be a well ordering. $M$ in $GL(n,R)$, Global orderings are well orderings (i.e., $1 < x_i$ for each variable $x_i$), local orderings satisfy $1 > x_i$ for each variable. If some variables are ordered globally and others locally we call it a mixed ordering. Local or mixed orderings are not well orderings.

Let $K$ be the ground field, $x = (x_1, \ldots, x_n)$ the variables and $<$ a monomial ordering, then Loc $K[x]$ denotes the localization of $K[x]$ with respect to the multiplicatively closed set

\begin{displaymath}\{1 +
g \mid g = 0 \hbox{ or } g \in K[x]\backslash \{0\} \hbox{ and }L(g) <
1\}.\end{displaymath}

Here, $L(g)$ denotes the leading monomial of $g$, i.e., the biggest monomial of $g$ with respect to $<$. The result of any computation which uses standard basis computations has to be interpreted in Loc $K[x]$. For all these orderings: Loc $K[x]$ = $K[x]$ $x^\alpha < x^\beta \Leftrightarrow \exists\; 1 \le i \le n :
\alpha_1 = \beta_1, \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i <
\beta_i$. $x^\alpha < x^\beta \Leftrightarrow \exists\; 1 \le i \le n :
\alpha_n = \beta_n,
\ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$ let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) < \deg(x^\beta)$ or $ \deg(x^\alpha) =
\deg(x^\beta)$ and $\exists\ 1 \le i \le n: \alpha_n = \beta_n,
\ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$ $ \deg(x^\alpha) =
\deg(x^\beta)$ and $\exists\ 1 \le i \le n:\alpha_1 = \beta_1,
\ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i < \beta_i.$ let $w_1, \ldots, w_n$ be positive integers. Then ${\tt wp}(w_1, \ldots,
w_n)$ $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$ let $w_1, \ldots, w_n$ be positive integers. Then ${\tt Wp}(w_1, \ldots,
w_n)$ Loc $K[x]$ = $K[x]_{(x)}$, $K[x]$ $(x_1, ..., x_n)$. $x^\alpha < x^\beta \Leftrightarrow \exists\; 1 \le i \le n :
\alpha_1 = \beta_1, \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i >
\beta_i$. $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) > \deg(x^\beta)$ or $ \deg(x^\alpha) =
\deg(x^\beta)$ and $\exists\ 1 \le i \le n: \alpha_n = \beta_n,
\ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$ $ \deg(x^\alpha) =
\deg(x^\beta)$ and $\exists\ 1 \le i \le n:\alpha_1 = \beta_1,
\ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i < \beta_i.$ ${\tt ws}(w_1, \ldots, w_n),\; w_1$ $w_2,\ldots,w_n$ ${\tt Ws}(w_1, \ldots, w_n),\; w_1$ $\{ x^a e_i \mid a \in N^n, 1 \leq i \leq r \}$ in Loc $K[x]^r$ = Loc $K[x]e_1
+ \ldots +$Loc $K[x]e_r$, where $e_1, \ldots, e_r$ denote the canonical generators of Loc $K[x]^r$, the r-fold direct sum of Loc $K[x]$. (The function gen(i) yields $e_i$). Loc $K[x]^r$ Loc $K[x]$ $<_m = (<,C)$ denotes the module ordering (giving priority to the coefficients):          $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow x^\alpha <
x^\beta$ or ( $x^\alpha = x^\beta $ and $ i < j$). $<_m = (C, <)$ denotes the module ordering (giving priority to the component):          $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow i < j$ or ($
i = j $ and $x^\alpha < x^\beta$). $<_m = (<,c)$ denotes the module ordering (giving priority to the coefficients):          $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow x^\alpha <
x^\beta$ or ( $x^\alpha = x^\beta $ and $ i > j$). $<_m = (c, <)$ denotes the module ordering (giving priority to the component):          $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow i > j$ or ($
i = j $ and $x^\alpha < x^\beta$). The output of a vector $v$ in $K[x]^r$ with components $v_1,
\ldots, v_r$ has the format $v_1 * gen(1) + \ldots + v_r * gen(r)$ In this case a vector is written as $[v_1, \ldots, v_r]$. $(n \times n)$-matrix $M_1, \ldots, M_n$ the rows of $M$.          $x^a < x^b \Leftrightarrow \exists\ 1 \leq i \leq n :
M_1 a = \; M_1 b, \ldots, M_{i-1} a = \; M_{i-1} b$ and $M_i a < \; M_i b$. $x^a < x^b$ if and only if $M a$ is smaller than $M b$

$\quad$ lp: $\left(\matrix{
1 & 0 & 0 \cr
0 & 1 & 0 \cr
0 & 0 & 1 \cr
}\right)$     dp: $\left(\matrix{
1 & 1 & 1 \cr
0 & 0 &-1 \cr
0 &-1 & 0 \cr
}\right)$     Dp: $\left(\matrix{
1 & 1 & 1 \cr
1 & 0 & 0 \cr
0 & 1 & 0 \cr
}\right)$

$\quad$ wp(1,2,3): $\left(\matrix{
1 & 2 & 3 \cr
0 & 0 &-1 \cr
0 &-1 & 0 \cr
}\right)$     Wp(1,2,3): $\left(\matrix{
1 & 2 & 3 \cr
1 & 0 & 0 \cr
0 & 1 & 0 \cr
}\right)$

$\quad$ ls: $\left(\matrix{
-1 & 0 & 0 \cr
0 &-1 & 0 \cr
0 & 0 &-1 \cr
}\right)$     ds: $\left(\matrix{
-1 &-1 &-1 \cr
0 & 0 &-1 \cr
0 &-1 & 0 \cr
}\right)$     Ds: $\left(\matrix{
-1 &-1 &-1 \cr
1 & 0 & 0 \cr
0 & 1 & 0 \cr
}\right)$

$\quad$ ws(1,2,3): $\left(\matrix{
-1 &-2 &-3 \cr
0 & 0 &-1 \cr
0 &-1 & 0 \cr
}\right)$     Ws(1,2,3): $\left(\matrix{
-1 &-2 &-3 \cr
1 & 0 & 0 \cr
0 & 1 & 0 \cr
}\right)$ $\quad$ (dp(3), wp(1,2,3)): $\left(\matrix{
1& 1& 1& 0& 0& 0 \cr
0& 0& -1& 0& 0& 0 \cr
0& -1& 0& 0& 0& 0 \cr
0& 0& 0& 1& 2& 3 \cr
0& 0& 0& 0& 0& -1 \cr
0& 0& 0& 0& -1& 0 \cr
}\right)$

$\quad$ (Dp(3), ds(3)): $\left(\matrix{
1& 1& 1& 0& 0& 0 \cr
1& 0& 0& 0& 0& 0 \cr
0& 1& 0& 0& 0& 0 \cr
0& 0& 0& -1& -1& -1 \cr
0& 0& 0& 0& 0& -1 \cr
0& 0& 0& 0& -1& 0 \cr
}\right)$ $\quad$ (dp(3), a(1,2,3),dp(3)): $\left(\matrix{
1& 1& 1& 0& 0& 0 \cr
0& 0& -1& 0& 0& 0 \cr
0& -1& 0& 0& 0& 0 \cr...
... \cr
0& 0& 0& 1& 1& 1 \cr
0& 0& 0& 0& 0& -1 \cr
0& 0& 0& 0& -1& 0 \cr
}\right)$

$\quad$ (a(1,2,3,4,5),Dp(3), ds(3)): $\left(\matrix{
1& 2& 3& 4& 5& 0 \cr
1& 1& 1& 0& 0& 0 \cr
1& 0& 0& 0& 0& 0 \cr
0...
...
0& 0& 0& -1& -1& -1 \cr
0& 0& 0& 0& 0 & -1 \cr
0& 0& 0& 0& -1& 0 \cr
}\right)$ $n \times n$ $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_m)$ $<_1$ a monomial ordering on $K[x]$ and $<_2$ a monomial ordering on $K[y]$. The product ordering (or block ordering) $<\ := (<_1,<_2)$ on $K[x,y]$ is the following:          $x^a y^b < x^A y^B \Leftrightarrow x^a <_1 x^A $ or ($x^a =
x^A$ and $y^b <_2 y^B$). ${\tt a}(w_1, \ldots, w_n),\; $ $w_1, \ldots, w_n$ $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n$

\begin{displaymath}\deg(x^\alpha) < \deg(x^\beta) \Rightarrow x^\alpha < x^\beta,\end{displaymath}


\begin{displaymath}\deg(x^\alpha) > \deg(x^\beta) \Rightarrow x^\alpha > x^\beta. \end{displaymath}

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 standard basis of $I$ if $L(f_1), \ldots, L(f_s)$ generate $L(I)$. A function $\hbox{NF} : R^r \times \{G \mid G\ \hbox{ a standard
basis}\} \to R^r, (p,G) \mapsto \hbox{NF}(p\vert G)$, is called a normal form if for any $p \in R^r$ and any standard basis $G$ the following holds: if $\hbox{NF}(p\vert G) \not= 0$ then $L(g)$ does not divide $L(\hbox{NF}(p\vert G))$ for all $g \in G$.

$\hbox{NF}(p\vert G)$ is called a normal form of $p$ with respect to $G$ (note that such a function is not unique). For a standard basis $G$ of $I$ the following holds: $f \in I$ if and only if $\hbox{NF}(f,G) = 0$. Let $I \subseteq K[\underline{x}]^r$ 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., $H_I=H_{L(I)}$. 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 Hilbert function of $M$, $H_M$, is defined (on the integers) by

\begin{displaymath}H_M(k) :=dim_K M_k.\end{displaymath}

The Hilbert-Poincare series of $M$ is the power series

\begin{displaymath}\hbox{HP}_M(t) :=\sum_{i=-\infty}^\infty
H_M(i)t^i=\sum_{i=-\infty}^\infty dim_K M_i \cdot t^i.\end{displaymath}

It turns out that $\hbox{HP}_M(t)$ can be written in two useful ways for weights $(1,..,1)$:

\begin{displaymath}\hbox{HP}_M(t)={Q(t)\over (1-t)^n}={P(t)\over (1-t)^{dim(M)}}\end{displaymath}

where $Q(t)$ and $P(t)$ are polynomials in ${\bf Z}[t]$. $Q(t)$ 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 $H_M(s)=\sum_{k=0}^N a_k$ ${d+s-k-1}\choose{d-1}$ (the Hilbert polynomial) for $s \ge N$. Generalizing these to quasihomogeneous modules we get

\begin{displaymath}\hbox{HP}_M(t)={Q(t)\over {\Pi_{i=1}^n(1-t^{w_i})}}\end{displaymath}

where $Q(t)$ is a polynomial in ${\bf Z}[t]$. $Q(t)$ is called the first (weighted) Hilbert series of M. Let $R$ be a quotient of $\hbox{Loc}_< K[\underline{x}]$ and let $I=(g_1, ..., g_s)$ be a submodule of $R^r$. Then the module of syzygies (or 1st syzygy module, module of relations) of $I$, syz($I$), is defined to be the kernel of the map $R^s \rightarrow R^r,\; \sum_{i=1}^s w_ie_i \mapsto \sum_{i=1}^s w_ig_i$. $(k-1)$-st 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. Let $I=(g_1,...,g_s)\subseteq R^r$ and $M= R^r/I$. A free resolution of $M$ is a long exact sequence

\begin{displaymath}...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F...
...1}\over{\longrightarrow} F_0\longrightarrow M\longrightarrow
0,\end{displaymath}

$R = \hbox{Loc}_< K[\underline{x}]$ 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

\begin{displaymath}
R^r = \bigoplus_a R\cdot e_{a,0} \buildrel A_1 \over \longl...
...ts \longleftarrow
\bigoplus_a R\cdot e_{a,n} \longleftarrow 0
\end{displaymath}

be a minimal free resolution of $R^n/I$ considered with homogeneous maps of degree 0. Then the 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 $I$. $s$

\begin{displaymath}
\hbox{deg}(e_{a,j}) \le s+j-1 \quad \hbox{for all $j$.}
\end{displaymath}

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 ini $(f):=a_s(x_1,...,x_{k-1})$. The pseudo remainder $r=\hbox{prem}(g,f)$ 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

\begin{displaymath}\hbox{lvar}(d_i)<\hbox{lvar}(f_i) =
\hbox{lvar}(f_i')=\hbox{lvar}(f_i''),\end{displaymath}


\begin{displaymath}0 \not\in \hbox{prem}(\{ d_i, \hbox{ini}(f_i'),
\hbox{ini}(f_i'')\},\{ f_1,...,f_{i-1}\}),\end{displaymath}


\begin{displaymath}\hbox{prem}(d_if_i-f_i'f_i'',\{f_1,...,f_{i-1}\})=0.\end{displaymath}

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 irreducible characteristic series of the ideal $(G)$. Let $f\colon(C^{n+1},0)\rightarrow(C,0)$ be a complex isolated hypersurface singularity given by a polynomial with algebraic coefficients which we also denote by $f$. Let $O=C[x_0,\ldots,x_n]_{(x_0,\ldots,x_n)}$ be the local ring at the origin and $J_f$ the Jacobian ideal of $f$.

A 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 $\partial_t$. The 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 moderate growth at $0$ form a regular $D=C\{t\}[\partial_t]$-module $G$, the 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 Brieskorn lattice $H'':=\Omega^{n+1}_{C^{n+1},0}/df\wedge d\Omega^{n-1}_{C^{n+1},0}$ in $G$.

The $D$-module structure defines the V-filtration $V$ on $G$ by $V^\alpha:=\sum_{\beta\ge\alpha}C\{t\}ker(t\partial_t-\beta)^{n+1}$. The Brieskorn lattice defines the Hodge filtration $F$ on $G$ by $F_k=\partial_t^kH''$ which comes from the mixed Hodge structure on the Milnor fibre. Note that $F_{-1}=H'$.

The induced V-filtration on the Brieskorn lattice determines the singularity spectrum $Sp$ by $Sp(\alpha):=\dim_CGr_V^\alpha Gr^F_0G$. The spectrum consists of $\mu$ rational numbers $\alpha_1,\dots,\alpha_\mu$ such that $e^{2\pi i\alpha_1},\dots,e^{2\pi i\alpha_\mu}$ are the eigenvalues of the monodromy. These 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 singularities of a specific type on a hypersurface $X\subset{P}^n$ 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 $x_0^d+\dots+x_n^d=0$, the singularities are bounded by the spectrum of $x_0^d+\dots+x_n^d$.

Using the library gaussman.lib one can compute the monodromy, the V-filtration on $H''/H'$, and the spectrum. $f=x^5+x^2y^2+y^5$ $\exp(2\pi iM)$ $H''/H'$ $O/J_f\cong H''/H'$ If the principal part of $f$ is $C$-nondegenerate, one can compute the spectrum using the library 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. $\tilde{E}_6$ on a surface $X\subset{P}^3$ $\mu(f)$, the geometric genus $p_g(f)$ $x^7+y^7+z^7=0$ ${3 \over 7}, {4 \over 7}, {5 \over 7}, {6 \over 7}, {1 \over 1},
{8 \over 7}, {...
...3 \over 7}, {2 \over 1}, {15 \over 7}, {16 \over 7}, {17 \over 7},
{18 \over 7}$ The singularities of type $\tilde{E}_6$ form a $\mu$-constant one parameter family given by $x^3+y^3+z^3+\lambda xyz=0,\quad \lambda^3\neq-27$. $x^3+y^3+z^3$. $\tilde{E}_6$ on a septic in $P^3$. But $x^7+y^7+z^7$ $\tilde{E}_6$. Let $A$ denote an $ m \times n $ matrix with integral coefficients. For $u
\in Z\!\!\! Z^n$, we define $u^+,u^-$ to be the uniquely determined vectors with nonnegative coefficients and disjoint support (i.e., $u_i^+=0$ or $u_i^-=0$ for each component $i$) such that $u=u^+-u^-$. For $u\geq 0$ component-wise, let $x^u$ denote the monomial $x_1^{u_1}\cdot\ldots\cdot x_n^{u_n}\in K[x_1,\ldots,x_n]$.

The ideal

\begin{displaymath}I_A:=<x^{u^+}-x^{u^-} \vert u\in\ker(A)\cap Z\!\!\! Z^n>\ \subset
K[x_1,\ldots,x_n] \end{displaymath}

is called a toric ideal.

The first problem in computing toric ideals is to find a finite generating set: Let $v_1,
\ldots, v_r$ be a lattice basis of $\ker(A)\cap
Z\!\!\! Z^n$ (i.e, a basis of the $Z\!\!\! Z$-module). Then

\begin{displaymath}I_A:=I:(x_1\cdot\ldots\cdot x_n)^\infty \end{displaymath}

where

\begin{displaymath}I=<x^{v_i^+}-x^{v_i^-}\vert i=1,\ldots,r> \end{displaymath}

section Algorithms. computes $I_A$ via the extended matrix $B=(I_m\vert A)$, where $I_m$ is the $m\times m$ unity matrix. A lattice basis of $B$ is given by the set of vectors $(a^j,-e_j)\in Z\!\!\! Z^{m+n}$, where $a^j$ is the $j$-th row of $A$ and $e_j$ the $j$-th coordinate vector. We look at the ideal in $K[y_1,\ldots,y_m,x_1,\ldots,x_n]$ corresponding to these vectors, namely

\begin{displaymath}I_1=<y^{a_j^+}- x_j y^{a_j^-} \vert j=1,\ldots, n>.\end{displaymath}

We introduce a further variable $t$ and adjoin the binomial $t\cdot
y_1\cdot\ldots\cdot y_m -1$ to the generating set of $I_1$, obtaining an ideal $I_2$ in the polynomial ring $K[t,
y_1,\ldots,y_m,x_1,\ldots,x_n]$. $I_2$ is saturated w.r.t. all variables because all variables are invertible modulo $I_2$. Now $I_A$ can be computed from $I_2$ by eliminating the variables $t,y_1,\ldots,y_m$. basis $v_1,
\ldots, v_r$ for the integer kernel of $A$ using the LLL-algorithm. The ideal corresponding to the lattice basis vectors

\begin{displaymath}I_1=<x^{v_i^+}-x^{v_i^-}\vert i=1,\ldots,r> \end{displaymath}

is saturated - as in the algorithm of Conti and Traverso - by inversion of all variables: One adds an auxiliary variable $t$ and the generator $t\cdot x_1\cdot\ldots\cdot x_n -1$ to obtain an ideal $I_2$ in $K[t,x_1,\ldots,x_n]$ from which one computes $I_A$ by elimination of $t$. compute $I_A$ without any auxiliary variables, provided that $A$ contains a vector $w$ with positive coefficients in its row space. This is a real restriction, i.e., the algorithm will not necessarily work in the general case.

A lattice basis $v_1,
\ldots, v_r$ is again computed via the LLL-algorithm. The saturation step is performed in the following way: First note that $w$ induces a positive grading w.r.t. which the ideal

\begin{displaymath}I=<x^{v_i^+}-x^{v_i^-}\vert i=1,\ldots,r> \end{displaymath}

corresponding to our lattice basis is homogeneous. We use the following lemma:

Let $I$ be a homogeneous ideal w.r.t. the weighted reverse lexicographical ordering with weight vector $w$ and variable order $x_1
> x_2 > \ldots > x_n$. Let $G$ denote a Groebner basis of $I$ w.r.t. to this ordering. Then a Groebner basis of $(I:x_n^\infty)$ is obtained by dividing each element of $G$ by the highest possible power of $x_n$.

From this fact, we can successively compute

\begin{displaymath}I_A= I:(x_1\cdot\ldots\cdot x_n)^\infty
=(((I:x_1^\infty):x_2^\infty):\ldots :x_n^\infty); \end{displaymath}

in the $i$-th step we take $x_i$ as the cheapest variable and apply the lemma with $x_i$ instead of $x_n$.

This procedure involves $n$ Groebner basis computations. Actually, this number can be reduced to at most $n/2$ to $n/2$ Groebner basis computations. It needs no auxiliary variables, but a supplementary precondition; namely, the existence of a vector without zero components in the kernel of $A$.

The main idea comes from the following observation:

Let $B$ be an integer matrix, $u_1,\ldots,u_r$ a lattice basis of the integer kernel of $B$. Assume that all components of $u_1$ are positive. Then

\begin{displaymath}I_B=<x^{u_i^+}-x^{u_i^-}\vert i=1,\ldots,r>, \end{displaymath}

i.e., the ideal on the right is already saturated w.r.t. all variables.

The algorithm starts by finding a lattice basis $v_1,
\ldots, v_r$ of the kernel of $A$ such that $v_1$ has no zero component. Let $\{i_1,\ldots,i_l\}$ be the set of indices $i$ with $v_{1,i}<0$. Multiplying the components $i_1,\ldots,i_l$ of $v_1,
\ldots, v_r$ and the columns $i_1,\ldots,i_l$ of $A$ by $-1$ yields a matrix $B$ and a lattice basis $u_1,\ldots,u_r$ of the kernel of $B$ that fulfill the assumption of the observation above. We are then able to compute a generating set of $I_A$ by applying the following ``variable flip'' successively to $i=i_1,\ldots,i_l$:

Let $>$ be an elimination ordering for $x_i$. Let $A_i$ be the matrix obtained by multiplying the $i$-th column of $A$ with $-1$. Let

\begin{displaymath}\{x_i^{r_j} x^{a_j} - x^{b_j} \vert j\in J \}\end{displaymath}

be a Groebner basis of $I_{A_i}$ w.r.t. $>$ (where $x_i$ is neither involved in $x^{a_j}$ nor in $x^{b_j}$). Then

\begin{displaymath}\{x^{a_j} - x_i^{r_j} x^{b_j} \vert j\in J \}\end{displaymath}

is a generating set for $I_A$. variable $u$ and one supplementary generator $x_1\cdot\ldots\cdot x_n -
u$ (instead of the generator $t\cdot x_1\cdot\ldots\cdot x_n -1$ in the algorithm of Pottier). The algorithm uses a quite unusual technique to get rid of the variable $u$ again. Let $A$ be an $ m \times n $ matrix with integral coefficients, $b\in
Z\!\!\! Z^m$ and $c\in Z\!\!\! Z^n$. The problem

\begin{displaymath}\min\{c^T x \vert x\in Z\!\!\! Z^n, Ax=b, x\geq 0\hbox{
component-wise}\} \end{displaymath}

is called an instance of the integer programming problem or IP problem.

The IP problem is very hard; namely, it is NP-complete.

For the following discussion let $c\geq 0$ (component-wise). We consider $c$ as a weight vector; because of its non-negativity, $c$ can be refined into a monomial ordering $>_c$. It turns out that we can solve such an IP instance with the help of toric ideals:

First we assume that an initial solution $v$ (i.e., $v\in Z\!\!\!
Z^n, v\geq 0, Av=b$) is already known. We obtain the optimal solution $v_0$ (i.e., with $c^T v_0$ minimal) by the following procedure: $>_c$ $x^v$ $x^(v_0)$ $v_0$ Faugère, $ 2*genus-2 < deg(G) < size(D) $ $\Omega(G-D)$ $epsilon$ $delta$ $epsilon + genus$ $epsilon:=[(deg(G)-3*genus+1)/2]$