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Next: 6. Some local algorithms Up: Computer Algebra and Algebraic Previous: 2. Normalisation   Contents

5. Singularities and standard bases

A (complex) singularity is, by definition, nothing but a complex analytic germ $ (V,0)$ together with its analytic local ring $ R = {\mathbb{C}}\{x\}/I$, where $ {\mathbb{C}}\{x\}$ is the convergent power series ring in $ x = x_1, \dots, x_n$. For an arbitrary field $ K$ let $ R = K[[x]] /I$ for some ideal $ I$ in the formal power series ring $ K[[x]]$. We call $ (V,0) = ($Spec$ \,R, \mathfrak{m})$ or just $ R$ a singularity ( $ \mathfrak{m}$ denotes the maximal ideal of the local ring $ R$) and write $ K\langle x \rangle$ for the convergent and for the formal power series ring if the statements hold for both.

If % latex2html id marker 5487
$ I \subset K[x]$ is an ideal with % latex2html id marker 5489
$ I \subset \langle x \rangle = \langle
x_1, \dots, x_n\rangle$ then the singularity of $ V(I)$ at $ 0 \in K^n$ is, using the above notation, $ K \langle x\rangle /I \cdot K\langle x\rangle $. However, we may also consider the local ring $ K[x]_{\langle x \rangle}/I \cdot
K[x]_{\langle x \rangle}$ with $ K[x]_{\langle x\rangle}$ the localisation of $ K[x]$ at $ \langle x \rangle$, as the singularity of $ V(I)$ at 0. Geometrically, for $ K = {\mathbb{C}}$, the difference is the following: $ {\mathbb{C}}\{x\}/I
{\mathbb{C}}\{x\}$ describes the variety $ V(I)$ in an arbitrary small neighbourhood of 0 in the Euclidean topology while $ {\mathbb{C}}[x]_{\langle x \rangle}/I
{\mathbb{C}}[x]_{\langle x \rangle}$ describes $ V(I)$ in an arbitrary small neighbourhood of 0 in the (much coarser) Zariski topology.

At the moment, we can compute efficiently only in $ K[x]_{\langle x\rangle}$ as we shall explain below. In many cases of interest, we are happy since invariants of $ V(I)$ at 0 can be computed in $ K[x]_{\langle x\rangle}$ as well as in $ K\langle x \rangle$. There are, however, others (such as factorisation), which are completely different in both rings.

$ (V,0)$ is called non-singular or regular or smooth if $ K\langle x \rangle/I$ is isomorphic (as local ring) to a power series ring $ K\langle y_1, \dots, y_d\rangle$, or if $ K[x]_{\langle x \rangle}/I$ is a regular local ring.

By the implicit function theorem, or by the Jacobian criterion, this is equivalent to the fact that $ I$ has a system of generators $ g_1, \dots,
g_{n-d}$ such that the Jacobian matrix of $ g_1, \dots,
g_{n-d}$ has rank $ n-d$ in some neighbourhood of 0. $ (V,0)$ is called an isolated singularity if there is a neighbourhood $ W$ of 0 such that $ W \cap (V
\smallsetminus\{0\})$ is regular everywhere.

In order to compute with singularities, we need the notion of standard basis which is a generalisation of the notion of Gröbner basis, cf. GP1,GP2.

A monomial ordering is a total order on the set of monomials $ \{x^\alpha \vert \alpha \in {\mathbb{N}}^n\}$ satisfying

$\displaystyle x^\alpha > x^\beta \Rightarrow x^{\alpha + \gamma} > x^{\beta + \gamma}$    for all $\displaystyle \alpha, \beta, \gamma \in {\mathbb{N}}^n.

We call a monomial ordering $ >$ global (resp. local, resp. mixed) if $ x_i > 1$ for all $ i$ (resp. $ x_i< 1$ for all $ i$, resp. if there exist $ i,j$ so that $ x_i< 1$ and $ x_j > 1$). This notion is justified by the associated ring to be defined below. Note that $ >$ is global if and only if $ >$ is a well-ordering (which is usually assumed).

Any $ f \in K[x]\smallsetminus\{0\}$ can be written uniquely as $ f = cx^\alpha
+ f'$, with $ c \in K\smallsetminus\{0\}$ and $ \alpha > \alpha'$ for any non-zero term $ c' x^{\alpha'}$ of $ f'$. We set lm $ (f) = x^\alpha$, the leading monomial of $ f$ and lc$ \,(f) = c$, the leading coefficient of $ f$.

For a subset % latex2html id marker 5598
$ G \subset K[x]$ we define the leading ideal of $ G$ as

$\displaystyle L(G) = \langle$    lm$\displaystyle (g) \;\vert\; g \in G\smallsetminus\{0\}\rangle_{K[x]},

the ideal generated by the leading monomials in $ G\smallsetminus\{0\}$.

So far, the general case is not different to the case of a well-ordering. However, the following definition provides something new for non-global orderings:

For a monomial ordering $ >$ define the multiplicatively closed set

$\displaystyle S_> := \{u \in K[x]\smallsetminus\{0\}\;\vert\;$lm$\displaystyle \,(u) = 1\}

and the $ K$-algebra

$\displaystyle R:=$   Loc$\displaystyle \,K[x] := S^{-1}_> K[x] = \{\dfrac{f}{u} \;\vert\; f \in K[x], u \in

the localisation (ring of fractions) of $ K[x]$ with respect to $ S_>$. We call Loc$ \,K[x]$ also the ring associated to $ K[x]$ and $ >$.

Note that % latex2html id marker 5627
$ K[x] \subset$   Loc% latex2html id marker 5628
$ \,K[x] \subset K[x]_{\langle x\rangle}$ and Loc$ \,
K[x] = K[x]$ if and only if $ >$ is global and Loc$ \,K[x] = K[x]_{\langle
x\rangle}$ if and only if $ >$ is local (which justifies the names).

Let $ >$ be a fixed monomial ordering. In order to have a short notation, I write

$\displaystyle R:=$   Loc$\displaystyle \,K[x] = S^{-1}_> K[x]

to denote the localisation of $ K[x]$ with respect to $ >$.

Let % latex2html id marker 5647
$ I \subset R$ be an ideal. A finite set % latex2html id marker 5649
$ G \subset I$ is called a standard basis of $ I$ if and only if $ L(G) = L(I)$, that is, for any $ f \in I\smallsetminus\{0\}$ there exists a $ g \in G$ satisfying lm$ \,(g)\vert$lm$ \,(f)$.

If the ordering is a well-ordering, then a standard basis $ G$ is called a Gröbner basis. In this case $ R = K[x]$ and, hence, % latex2html id marker 5666
$ G
\subset I \subset K[x]$.

Standard bases can be computed in the same way as Gröbner bases except that we need a different normal form. This was first noticed by Mo for local orderings (called tangent cone orderings by Mora) and, in general, by GP1,Getal.

Let $ {\mathcal G}$ denote the set of all finite and ordered subsets % latex2html id marker 5670
$ G \subset R$. A map

   NF$\displaystyle \,: R \times {\mathcal G}\to R,\; (f,G) \mapsto$   NF$\displaystyle \,(f\vert G),

is called a normal form on $ R$ if, for all $ f$ and $ G$,

NF$ \,(f\vert G) \not= 0 \Rightarrow$   lm$ \,\bigl($NF$ \,(f\vert G)\bigr) \not\in
$ f -$   NF$ \,(f\vert G) \in \langle G\rangle_R$, the ideal in $ R$ generated by $ G$.

NF is called a weak normal form if, instead of (ii), only the following condition (ii') holds:

for each $ f \in R$ and each $ G \in {\mathcal G}$ there exists a unit $ u
\in R$, so that $ uf-$NF$ \,(f\vert G) \in \langle G\rangle_R$.

Moreover, we need (in particular for computing syzygies) (weak) normal forms with standard representation: if $ G = \{g_1, \dots, g_k\}$, we can write

$\displaystyle f -$   NF$\displaystyle \,(f\vert G) = \sum^k_{i=1} a_i g_i,\quad a_i \in R,

such that lm$ \,\bigl(f-$NF$ \,(f\vert G)\bigr) \ge$   lm$ \,(a_ig_i)$ for all $ i$, that is, no cancellation of bigger leading terms occurs among the $ a_i g_i$.

Indeed, if $ f$ and $ G$ consist of polynomials, we can compute, in finitely many steps, weak normal forms with standard representation such that $ u$ and NF$ \,(f\vert G)$ are polynomials and, hence, compute polynomial standard bases which enjoy most of the properties of Gröbner bases.

Once we have a weak normal form with standard representation, the general standard basis algorithm may be formalised as follows:

STANDARDBASIS(G,NF) [arbitrary monomial ordering]

Input: $ G$ a finite and ordered set of polynomials, NF a weak normal form with standard representation.

Output: $ S$ a finite set of polynomials which is a standard basis of $ \langle G\rangle_R$.

- $ S = G$;
- $ P = \{(f,g) \mid f,g\in S\}$;
- while $ (P \not= \emptyset)$
choose $ (f,g) \in P$;
$ P = P \smallsetminus \{(f,g)\}$;
$ h =$   NF$ \,($spoly$ (f,g)\mid S)$;
if $ (h \not= 0)$
$ P = P \cup \{(h,f) \mid f \in S\}$;
$ S = S \cup \{h\}$;
- return $ S$;

Here spoly $ (f,g) = x^{\gamma - \alpha} f - \tfrac{lc(f)}{lc(g)} x^{\gamma -
\beta} g$ denotes the $ \boldsymbol{s}$-polynomial of $ f$ and $ g$ where $ x^\alpha =$   lm$ \,(f),\; x^\beta =$   lm$ \,(g),\; \gamma = lcm(\alpha,\beta)$.

The algorithm terminates by Dickson's lemma or by the noetherian property of the polynomial ring (and since NF terminates). It is correct by Buchberger's criterion, which generalises to non-well-orderings.

If we use Buchberger's normal form below, in the case of a well-ordering, STANDARDBASIS ist just Buchberger's algorithm:

NFBUCHBERGER(f,G) [well-ordering]

Input: $ G$ a finite ordered set of polynomials, $ f$ a polynomial.

Output: $ h$ a normal form of $ f$ with respect to $ G$ with standard representation.

- $ h = f$;
- while $ (h \not= 0$ and exist $ g \in G$ so that lm$ \,(g)\mid$lm$ \,(h))$
choose any such $ g$;
$ h =$   spoly$ (h,g)$;
- return $ h$;

For an algorithm to compute a weak normal form in the case of an arbitrary ordering, we refer to GP1.

To illustrate the difference between local and global orderings, we compute the dimension of a variety at a point and the (global) dimension of the variety.

The dimension of the singularity $ (V,0)$, or the dimension of $ V$ at 0, is, by definition, the Krull dimension of the analytic local ring $ {\mathcal O}_{V,0} = K\langle x \rangle/I$, which is the same as the Krull dimension of the algebraic local ring $ K[x]_{\langle x \rangle}/I$ in case $ I = \langle f_1,
\dots, f_k\rangle$ is generated by polynomials, which follows easily from the theory of dimensions by Hilbert-Samuel series.

Using this fact, we can compute $ \dim(V,0)$ by computing a standard basis of the ideal $ \langle f_1, \dots, f_k\rangle$ generated in Loc$ \,K[x]$ with respect to any local monomial ordering on $ K[x]$. The dimension is equal to the dimension of the corresponding monomial ideal (which is a combinatorial problem).

For example, the dimension of the affine variety $ V = V(yx-y, zx-z)$ is 2 but the dimension of the singularity $ (V,0)$ (that is, the dimension of $ V$ at the point 0) is 1:


$ V : y(x-1) = z(x-1) = 0$,
$ \dim(V,0) = 1,\; \dim V = 2$

Using SINGULAR we compute first the global dimension with the degree reverse lexicographical ordering denoted by dp and then the local dimension at 0 using the negative degree reverse lexicographical ordering denoted by ds. Note that in the local ring $ K[x,y]_{\langle x,y\rangle}$ (represented by the ordering ds) $ x-1$ is a unit.

ring R   = 0,(x,y,z),dp;  //global ring
ideal i  = yx-y,zx-z;
ideal si = groebner(i);
==> si[1]=xz-z,           //leading ideal of i is <xz,xy>  
==> si[2]=xy-y
==> 2                     //global dimension = dim R/<xz,xy>

ring r   = 0,(x,y,z),ds;  //local ring
ideal i  = yx-y,zx-z;
ideal si = groebner(i);
==> si[1]=y               //leading ideal of i is <y,z>
==> si[2]=z
==> 1                     //local dimension = dim r/<y,z>

next up previous contents
Next: 6. Some local algorithms Up: Computer Algebra and Algebraic Previous: 2. Normalisation   Contents
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