3.3.2 General syntax of a ring declaration

Rings

Syntax:

ring name = (coefficients), ( names_of_ring_variables ), ( ordering ); or
ring name = cring [ names_of_ring_variables ]

Default:

(ZZ/32003)[x,y,z];

Purpose:

declares a ring and sets it as the current basering. The second form sets the ordering to (dp,C). cring stands currently for QQ (the rationals), ZZ (the integers) or (ZZ/m) (the field (m prime and <2147483648) resp. ring of the integers modulo m).

The coefficients (for the first form) are given by one of the following:

1. a cring as given above
2. a non-negative int_expression less than 2147483648 (2^31).
   The int_expression should either be 0, specifying the field of rational numbers Q, or a prime number p, specifying the finite field with p elements. If it is not a prime number, int_expression is converted to the next lower prime number.

3. an expression_list of an int_expression and one or more names.
   The int_expression specifies the characteristic of the coefficient field as described above. The names are used as parameters in transcendental or algebraic extensions of the coefficient field. Algebraic extensions are implemented for one parameter only. In this case, a minimal polynomial has to be defined by an assignment to minpoly. See minpoly.

4. an expression_list of an int_expression and a name.
   The int_expression has to be a prime number p to the power of a positive integer n. This defines the Galois field with elements, where has to be less than or equal to .The given name refers to a primitive element of generating the multiplicative group. Due to a different internal representation, the arithmetic operations in these coefficient fields are faster than arithmetic operations in algebraic extensions as described above.

5. an expression_list of the name real and two optional int_expressions determining the precision in decimal digits and the size for the stabilizing rest. The default for the rest is the same size as for the representation. An exception is the name real without any integers. These numbers are implemented as machine floating point numbers of single precision. Note that computations over all these fields are not exact.

6. an expression_list of the name complex, two optional int_expression and a name. This specifies the field of complex numbers represented by floating point numbers with a precision similar to real. An expression_list without int_expression defines a precision and rest with length 6. The name of the imaginary unit is given by the last parameter. Note that computations over these fields are not exact.

7. an expression_list with the name integer. This specifies the ring of integers.

8. an expression_list with the name integer and one subsequent int_expression. This specifies the ring of integers modulo the given int_expression.

9. an expression_list with the name integer and two int_expressions b and e. This specifies the ring of integers modulo b^e. If b = 2 and e < int_bit_size an optimized implementation is used.

'names_of_ring_variables' is a list of names or indexed names.

'ordering' is a list of block orderings where each block ordering is either

1. lp, dp, Dp, ls, ds, or Ds optionally followed by a size parameter in parentheses.

2. wp, Wp, ws, Ws, or a followed by a weight vector given as an intvec_expression in parentheses.

3. M followed by an intmat_expression in parentheses.

4. c or C.

For the definition of the orderings, see Monomial orderings.

If one of coefficients, names_of_ring_variables, and ordering consists of only one entry, the parentheses around this entry may be omitted.

Quotient rings

Syntax:

qring name = ideal_expression ;

Default:

none

Purpose:

declares a quotient ring as the basering modulo ideal_expression, and sets it as current basering.

ideal_expression has to be represented by a standard basis.

The most convenient way to map objects from a ring to its quotient ring and vice versa is to use the fetch function (see fetch).

SINGULAR computes in a quotient ring as long as possible with the given representative of a polynomial, say, f. I.e., it usually does not reduce f w.r.t. the quotient ideal. This is only done when necessary during standard bases computations or by an explicit reduction using the command reduce(f, std(0)) (see reduce).

Operations based on standard bases (e.g. std,groebner, etc., reduce) and functions which require a standard basis (e.g. dim,hilb, etc.) operated with the residue classes; all others on the polynomial objects.

Example:

// definition and usage: ring r=(ZZ/32003)[x,y]; poly f=x3+yx2+3y+4; qring q=std(maxideal(2)); basering; ==> // coefficients: ZZ/32003 ==> // number of vars : 2 ==> // block 1 : ordering dp ==> // : names x y ==> // block 2 : ordering C ==> // quotient ring from ideal ==> _[1]=y2 ==> _[2]=xy ==> _[3]=x2 poly g=fetch(r, f); g; ==> x3+x2y+3y+4 reduce(g,std(0)); ==> 3y+4 // polynomial and residue class: ring R=QQ[x,y]; qring Q=std(y); poly p1=x; poly p2=x+y; // comparing polynomial objects: p1==p2; ==> 0 // comparing residue classes: reduce(p1,std(0))==reduce(p2,std(0)); ==> 1