
6.4 Miscellaneous oddities
 integer division
If two numerical constants (i.e., two sequences of digits) are divided
using the / operator, the surrounding whitespace determines
which division to use: if there is no space between the constants and
the / operator (e.g., "3/2"), both numerical constants are
treated as of type number and the current ring division is
used. If there is at least one space surrounding the / operator
(e.g., "3 / 2"), both numerical constants are treated as of type
int and an integer division is performed. To avoid confusion, use
the div operator instead of / for integer division and an
explicit type cast to number for ring division. Note, that this
problem does only occur for divisions of numerical constants.
It also apllies for large numerical constants which are of type bigint .
 ring r=32002,x,dp;
==> // ** 32002 is invalid as characteristic of the ground field. 32003 is us\
ed.
3/2; // ring division
==> 16000
3 / 2; // integer division
==> // ** int division with `/`: use `div` instead in line >> 3 / 2; // int\
eger division<<
==> 1
3 div 2;
==> 1
number(3) / number(2);
==> 16000
number a=3;
number b=2;
a/b;
==> 16000
int c=3;
int d=2;
c / d;
==> // ** int division with `/`: use `div` instead in line >> c / d;<<
==> 1

 monomials and precedence
The formation of a monomial has precedence over all operators
(a monomial is here an optional coefficient followed by any sequence
of ring variables (possibly followed by
an exponent) which only conssist of letters, digits
and (over the rationals) / without any whitespace):
 ring r=0,(x,y),dp;
2xy^2 == (2*x*y)^2;
==> 1
2xy^2 == 2x*y^2;
==> 0
2x*y^2 == 2*x * (y^2);
==> 1
 During that formation no operator is involved: in the noncommutative
case, we have
 LIB "nctools.lib";
ring r = 0,(x,y),dp;
def S = superCommutative();
xy == yx;
==> 1
x*y == y*x;
==> 1
x*y, y*x;
==> xy xy

 meaning of
mult
For an arbitrary ideal or module i , mult(i) returns the
multiplicity of the ideal generated by the leading monomials of the
given generators of i , hence depends on the monomial ordering!
A standard mistake is to interpret degree(i) or mult(i)
for an inhomogeneous ideal i as the degree of the homogenization
or as something like the 'degree of the affine part'. For the ordering
dp (degree reverse lexicographical) the converse is true: if
i is given by a standard basis, mult(i) is the degree of
the homogeneous ideal obtained by homogenization of i and then
putting the homogenizing variable to 0, hence it is the degree of the
part at infinity (this can also be checked by looking at the initial
ideal).
 size of ideals
size counts the nonzero entries of an ideal or module. Use
ncols to determine the actual number of entries in the ideal or module.
 computations in
qring
In order to speed up computations in quotient rings, SINGULAR
usually does not reduce polynomials w.r.t. the quotient ideal; rather
the given representative is used as long as possible during
computations. If it is necessary, reduction is done during standard base
computations. To reduce a polynomial f by hand w.r.t. the
current quotient ideal use the command reduce(f,std(0))
(see reduce).
 degree of a polynomial
degBound
The exact meaning of "degree" depends on the ring odering and the command:
slimgb uses always the total degree with weights 1,
std does so only for block orderings.
hilb
the degree is the total degree with weights 1 unless a weight vector is given
kbase
the degree is the total degree with weights 1
(to use another weight vector see weightKB)
 substring selection
To extract substrings from a string , square brackets are used,
enclosing either two commaseparated int s or an
intvec . Although two commaseparated int s represent an
intvec , they mean different things in substring access. Square
brackets enclosing two int s (e.g. s[2,6] ) return a
substring where the first integer denotes the starting position and the
second integer denotes the length of the substring. The result is
returned as a string . Square brackets enclosing an intvec
(e.g. s[intvec(2,6)] ) return the characters of the string at the
position given by the values of the intvec . The result is
returned as an expression list of strings.
 string s = "oneword";
s[2,6]; // a substring starting at the second char
==> newor
size(_);
==> 6
intvec v = 2,6;
s[v]; // the second and the sixth char
==> n o
string st = s[v]; // stick together by an assignment
st;
==> no
size(_);
==> 2
v = 2,6,8;
s[v];
==> n o d

 packages and indexed variables
See example
 package K;
string K::varok;
string K::donotwork(1);
int K::i(1..3);
// Toplevel does not contain i(1..3)
listvar();
==> // i(3) [0] int 0
==> // i(2) [0] int 0
==> // i(1) [0] int 0
==> // donotwork(1) [0] string
// i(1..3) are stored in Package 'K'
listvar(K);
==> // K [0] package K (N)
==> // ::varok [0] string

