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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 ordering has to be chosen such that the unit-elements of the ring are precisely those 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 from the user. The leading part represents the number and the rest is for the numerical stability. Two numbers with a difference only in the rest are equal.

  • the ring Z∕32003[x,y,z] with degree reverse lexicographical ordering. The exact ring declaration may be omitted in the first example since this is the default ring:
    ring r;
    ring r = 32003,(x,y,z),dp;
    
  • the ring Q[a,b,c,d] with lexicographical ordering:
    ring r = 0,(a,b,c,d),lp;
    
  • the ring Z∕7[x,y,z] with local degree reverse lexicographical ordering. The non-prime 10 is converted to the next lower prime in the second example:
    ring r = 7,(x,y,z),ds;
    ring r = 10,(x,y,z),ds;
    
  • the ring Z∕7[x1,,x6] with lexicographical ordering for x1,x2,x3 and degree reverse lexicographical ordering for x4,x5,x6:
    ring r = 7,(x(1..6)),(lp(3),dp);
    
  • the localization of (Q[a,b,c])[x,y,z] at the maximal ideal (x,y,z)

    :

    ring r = 0,(x,y,z,a,b,c),(ds(3), dp(3));
    
  • the ring Q[x,y,z] with weighted reverse lexicographical ordering. The variables x , y , and z have the weights 2, 1, and 3, respectively, and vectors are first ordered by components (in descending order) and then by monomials:
    ring r = 0,(x,y,z),(c,wp(2,1,3));
    

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

  • the ring K[x,y,z] , where K = Z∕7(a,b,c) denotes the transcendental extension of Z∕7 by a , b and c with degree lexicographical ordering:
    ring r = (7,a,b,c),(x,y,z),Dp;
    
  • the ring K[x,y,z] , where K = Z∕7[a] denotes the algebraic extension of degree 2 of Z∕7 by a. In other words, K is the finite field with 49 elements. In the first case, a denotes an algebraic element over Z∕7 with minimal polynomial μa = a2 + a + 3, in the second case, a

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

    ring r = (7,a),(x,y,z),dp; minpoly = a^2+a+3;
    ring r = (7^2,a),(x,y,z),dp;
    
  • the ring R[x,y,z] , where R 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 R[x,y,z] , where R 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 R[x,y,z] , where R 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 R(j)[x,y,z] , where R denotes the field of real numbers represented by floating point numbers of 30 valid decimal digits and the same number for the rest. j denotes the imaginary unit.
    ring r = (complex,30,j),(x,y,z),dp;
    
  • the ring R(i)[x,y,z] , where R denotes the field of real numbers represented by floating point numbers of 6 valid decimal digits and the same number for the rest. i is the default for the imaginary unit.
    ring r = complex,(x,y,z),dp;
    
  • the quotient ring Z∕7[x,y,z] modulo the square of the maximal ideal (x,y,z) :
    ring R = 7,(x,y,z), dp;
    qring r = std(maxideal(2));
    

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            User manual for Singular version 2-0-4, October 2002, generated by texinfo.