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
$Z/7[x_1,\ldots,x_6]$
with lexicographical ordering for
$x_1,x_2,x_3$
and degree reverse lexicographical ordering for
$x_4,x_5,x_6$:

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
$\mu_a=a^2+a+3$,
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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