### 3.3.1 Examples of ring declarations

The exact syntax of a ring declaration is given in the next two subsections; this subsection lists some examples first. Note that the chosen ordering implies that a unit-elements of the ring will be among the elements with leading monomial 1. For more information, see Monomial orderings.

Every floating point number in a ring consists of two parts, which may be chosen by the user. The leading part represents the number and the rest is for numerical stability. Two numbers with a difference only in the rest will be regarded equal.

- the ring with degree reverse lexicographical ordering. The exact ring declaration may be omitted in the first example since this is the default ring:

      ring r1;
      ring r2 = 32003,(x,y,z),dp;
      ring r3=(ZZ/32003)[x,y,z];
      ring r4 = (ZZ/32003),(x,y,z),dp;

- similar examples with indexed variables. The ring variables of r1 are going to be x(1)..x(10); in r2 they will be x(1)(1), x(1)(2), ..., x(1)(8), x(2)(1), ..., x(5)(8):

      ring r1 = 32003,(x(1..10)),dp;
      ring r2 = 32003,(x(1..5)(1..8)),dp;
      ring r3 = (ZZ/32003)[x(1..5)(1..8)];
      ring r4 = (ZZ/32003),(x(1..5)(1..8)),dp;

- the ring with lexicographical ordering:

      ring r1 = 0,(a,b,c,d),lp;
      ring r2 = QQ,(a,b,c,d),lp;

- the ring with local degree reverse lexicographical ordering. The non-prime 10 is converted to the next lower prime in the second example:

      ring r1 = 7,(x,y,z),ds;
      ring r2 = 10,(x,y,z),ds;
      ring r3 = (ZZ/7),(x,y,z),ds;

- the ring with lexicographical ordering for and degree reverse lexicographical ordering for :

      ring r1 = 7,(x(1..6)),(lp(3),dp);
      ring r2 = (ZZ/7),(x(1..6)),(lp(3),dp);

- the localization of at the maximal ideal

  :

      ring r1 = 0,(x,y,z,a,b,c),(ds(3), dp(3));
      ring r2 = QQ,(x,y,z,a,b,c),(ds(3), dp(3));

- the ring with weighted reverse lexicographical ordering. The variables , , and have the weights 2, 1, and 3, respectively, and vectors are first ordered by components (in descending order) and then by monomials:

      ring r1 = 0,(x,y,z),(c,wp(2,1,3));
      ring r2 = QQ,(x,y,z),(c,wp(2,1,3));

  For ascending component order, the component ordering `C` has to be used.

- the ring , where denotes the transcendental extension of by , and with degree lexicographical ordering:

      ring r = (7,a,b,c),(x,y,z),Dp;

- the ring , where denotes the algebraic extension of degree 2 of by In other words, is the finite field with 49 elements. In the first case, denotes an algebraic element over with minimal polynomial ,in the second case,

  refers to some generator of the cyclic group of units of :

      ring r = (7,a),(x,y,z),dp; minpoly = a^2+a+3;
      ring r = (7^2,a),(x,y,z),dp;

- the ring , where denotes the field of real numbers represented by simple precision floating point numbers. This is a special case:

      ring r = real,(x,y,z),dp;

- the ring , where denotes the field of real numbers represented by floating point numbers of 50 valid decimal digits and the same number of digits for the rest:

      ring r = (real,50),(x,y,z),dp;

- the ring , where denotes the field of real numbers represented by floating point numbers of 10 valid decimal digits and with 50 digits for the rest:

      ring r = (real,10,50),(x,y,z),dp;

- the ring , where denotes the field of real numbers represented by floating point numbers of 30 valid decimal digits and the same number for the rest. denotes the imaginary unit.

      ring r = (complex,30,j),(x,y,z),dp;

- the ring , where denotes the field of real numbers represented by floating point numbers of 6 valid decimal digits and the same number for the rest. is the default for the imaginary unit.

      ring r = complex,(x,y,z),dp;

- the quotient ring modulo the square of the maximal ideal :

      ring R = 7,(x,y,z), dp;
      qring r = std(maxideal(2));

- the ring :

      ring R = integer,(x,y,z), dp;

- the ring :

      ring R = (integer, 6, 3),(x,y,z), dp;

- the ring :

      ring R = (integer, 100),(x,y,z), dp;

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