To calculate with objects as ideals, matrices, modules, and polynomial vectors, a ring has to be defined first.
ring r = 0,(x,y,z),dp;
The definition of a ring consists of three parts: the first part
determines the ground field, the second part determines the names of the
ring variables, and the third part determines the monomial ordering to
be used. So the example above declares a polynomial ring called r
with a ground field of characteristic 0 (i.e., the rational
numbers) and ring variables called x
, y
, and z
. The
dp
at the end means that the degree reverse lexicographical
ordering should be used.
Other ring declarations:
ring r1=32003,(x,y,z),dp;
x
, y
, and z
and
ordering dp
.
ring r2=32003,(a,b,c,d),lp;
a
, b
, c
,
d
and lexicographical ordering.
ring r3=7,(x(1..10)),ds;
x(1)
,...,x(10)
, negative
degree revers lexicographical ordering (ds
).
ring r4=(0,a),(mu,nu),lp;
mu
and nu
.
Typing the name of a ring prints its definition. The example below shows, that the default ring in SINGULAR is Z/32003[x,y,z] with degree reverse lexicographical ordering:
ring r5; r5; ==> // characteristic : 32003 ==> // number of vars : 3 ==> // block 1 : ordering dp ==> // : names x y z ==> // block 2 : ordering C
Defining a ring makes this ring the current active basering, so each ring definition above switches to a new basering. The concept of rings in SINGULAR is discussed in detail in the chapter "Rings and orderings" of the SINGULAR manual.
The basering now is r5
. Since we want to calcualate in the ring
r
, which we defined first, we have to switch back to it. This can
be done using the function setring
:
setring r;
Once a ring is active, we can define polynomials. A monomial, say
x^3
may be entered in two ways: either using the power operator ^
,
saying x^3
, or in short-hand notation without operator, saying
x3
. Note, that the short-hand notation is forbidden if the name
of the ring variable consists of more than one character. Note, that
SINGULAR always expands brackets and automatically sorts the terms
with respect to the monomial ordering of the basering.
poly f = x3+y3+(x-y)*x2y2+z2; f; ==> x3y2-x2y3+x3+y3+z2
The command size
determines in general the number of "single
entries" in an object. In particular, for polynomials, size
determines the number of monomials.
size(f); ==> 5
A natural question is to ask if a point e.g. (x,y,z)=(1,2,0)
lies
on the variety defined by the polynomials f
and g
. For
this we define an ideal generated by both polynomials, substitute the
coordinates of the point for the ring variables, and check if the result
is zero:
poly g = f^2 *(2x-y); ideal I = f,g; ideal J= subst(I,var(1),1); J = subst(J,var(2),2); J = subst(J,var(3),0); J; ==> J[1]=5 ==> J[2]=0
Since the result is not zero, the point (1,2,0)
does
not lye on the variety V(f,g)
.
Another question is to decide whether some function vanishes on a
variety, or in algebraic terms if a polynomial is contained in a given
ideal. For this we calculate a standard basis using the command
groebner
and afterwards reduce the polynomial with respect to
this standard basis.
ideal sI = groebner(f); reduce(g,sI); ==> 0
As the result is 0
the polynomial g
belongs to the
ideal defined by f
.
The function groebner
, like many other functions in
SINGULAR, prints a protocol during calculation, if desired. The
command option(prot);
enables protocoling whereas
option(noprot);
turns it off.
The command kbase
calculates a basis of the polynomial ring
modulo an ideal, if the quotient ring is finite dimensional.
As an example we calculate the Milnor number of a
hypersurface singularity in the global and local case. This is the
vector space dimension of the polynomial ring modulo the Jacobian ideal
in the global case resp. of the power series ring modulo the Jacobian
ideal in the local case. See Section 4.3 Critical points, for a detailed
explanation.
The Jacobian ideal is obtained with the command jacob
.
ideal J = jacob(f); ==> // ** redefining J ** J; ==> J[1]=3x2y2-2xy3+3x2 ==> J[2]=2x3y-3x2y2+3y2 ==> J[3]=2z
SINGULAR prints the line // ** redefining J
**
. This indicates that we have previously defined a variable with name
J
of type ideal (see above).
To obtain a representing set of the quotient vectorspace we first
calculate a standard basis, then we apply the function kbase
to
this standard basis.
J = groebner(J); ideal K = kbase(J); K; ==> K[1]=y4 ==> K[2]=xy3 ==> K[3]=y3 ==> K[4]=xy2 ==> K[5]=y2 ==> K[6]=x2y ==> K[7]=xy ==> K[8]=y ==> K[9]=x3 ==> K[10]=x2 ==> K[11]=x ==> K[12]=1
Then
size(K); ==> 12
gives the desired vector space dimension K[x,y,z]/jacob(f). As in SINGULAR the functions may take the input directly from earlier calculations, the whole sequence of commands may be written in one single statement.
size(kbase(groebner(jacob(f)))); ==> 12
When we are not interested in a basis of the quotient vector space, but
only in the resulting dimension we may even use the command vdim
and write:
vdim(groebner(jacob(f))); ==> 12