
A.1.1 Basic programming
We show in the example below the following:

define the ring
R of characteristic 32003, variables x,y,z , monomial
ordering dp (implementing F_32003[x,y,z])

list the information about
R by typing its name

check the order of the variables

define the integers
a,b,c,t

define a polynomial
f (depending on a,b,c,t ) and display it

define the jacobian ideal
i of f

compute a Groebner basis of
i

compute the dimension of the algebraic set defined by
i (requires
the computation of a Groebner basis)

create and display a string in order to comment the result (text between
quotes " "; is a 'string')

load a library (see primdec_lib)

compute a primary decomposition for
i and assign the result to a
list L (which is a list of lists of ideals)

display the number of primary components and the first primary and prime
components (entries of the list L[1])

implement the localization of F_32003[x,y,z] at the homogeneous maximal
ideal (generated by x,y,z) by defining a ring with local monomial
ordering (
ds in place of dp )

map i to this ring (see imap)  we may use the same name
i ,
since ideals are ring dependent data

compute the local dimension of the algebraic set defined by
i at
the origin (= dimension of the ideal generated by i in the localization)

compute the local dimension of the algebraic set defined by
i at
the point (2000,6961,7944) (by applying a linear coordinate transformation)
For a more basic introduction to programming in SINGULAR, we refer
to Getting started.
 ring R = 32003,(x,y,z),dp;
R;
==> // characteristic : 32003
==> // number of vars : 3
==> // block 1 : ordering dp
==> // : names x y z
==> // block 2 : ordering C
x > y;
==> 1
y > z;
==> 1
int a,b,c,t = 1,2,1,4;
poly f = a*x3+b*xy3c*xz3+t*xy2z2;
f;
==> 4xy2z2+2xy3+xz3+x3
ideal i = jacob(f); // Jacobian Ideal of f
ideal si = std(i); // compute Groebner basis
int dimi = dim(si);
string s = "The dimension of V(i) is "+string(dimi)+".";
s;
==> The dimension of V(i) is 1.
LIB "primdec.lib"; // load library primdec.lib
list L = primdecGTZ(i);
size(L); // number of prime components
==> 6
L[1][1]; // first primary component
==> _[1]=2y2z2+y316001z3
==> _[2]=x
L[1][2]; // corresponding prime component
==> _[1]=2y2z2+y316001z3
==> _[2]=x
ring Rloc = 32003,(x,y,z),ds; // ds = local monomial ordering
ideal i = imap(R,i);
dim(std(i));
==> 1
map phi = R, x2000, y6961, z7944;
dim(std(phi(i)));
==> 0

