6.4 Miscellaneous oddities

1. 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 applies 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

2. 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 non-commutative 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

3. 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).

4. size of ideals

   size counts the non-zero entries of an ideal or module. Use ncols to determine the actual number of entries in the ideal or module.

5. 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).

6. 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)

7. substring selection

   To extract substrings from a string, square brackets are used, enclosing either two comma-separated ints or an intvec. Although two comma-separated ints represent an intvec, they mean different things in substring access. Square brackets enclosing two ints (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 = "one-word"; s[2,6]; // a substring starting at the second char ==> ne-wor 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

8. packages and indexed variables

   See example

   package K; string varok; exportto(K,varok); string work(1); exportto(K,work(1)); int i(1..3); exportto(K,i(1..3)); // Toplevel does not contain i(1..3) listvar(); // i(1..3) are stored in Package 'K' listvar(K); ==> // K [0] package K (N) ==> // ::i(3) [0] int 0 ==> // ::i(2) [0] int 0 ==> // ::i(1) [0] int 0 ==> // ::work(1) [0] string ==> // ::varok [0] string