Singular Manual: Rings associated to monomial orderings

| ### A.1.3 Rings associated to monomial orderings In we may implement localizations of the polynomial ring by choosing an appropriate monomial ordering (when defining the ring by the `ring` command). We refer to Monomial orderings for a thorough discussion of the monomial orderings available in . At this point, we restrict ourselves to describing the relation between a monomial ordering and the ring (as mathematical object) which is implemented by the ordering. This is most easily done by describing the set of units: if > is a monomial ordering then precisely those elements which have leading monomial 1 are considered as units (in all computations performed with respect to this ordering). In mathematical terms: choosing a monomial ordering `>` implements the localization of the polynomial ring with respect to the multiplicatively closed set of polynomials with leading monomial 1. If > is global (that is, 1 is the smallest monomial), the implemented ring is just the polynomial ring. If > is local (that is, if 1 is the largest monomial), the implemented ring is the localization of the polynomial ring w.r.t. the homogeneous maximal ideal. For a mixed ordering, we obtain "something in between these two rings":
    ring R = 0,(x,y,z),dp;     // polynomial ring (global ordering)
    poly f = y4z3+2x2y2z2+4z4+5y2+1;
    f;                         // display f in a degrevlex-ordered way
    ==> y4z3+2x2y2z2+4z4+5y2+1
    short=0;                   // avoid short notation
    f;
    ==> y^4*z^3+2*x^2*y^2*z^2+4*z^4+5*y^2+1
    short=1;
    leadmonom(f);              // leading monomial
    ==> y4z3

    ring r = 0,(x,y,z),ds;     // local ring (local ordering)
    poly f = fetch(R,f);
    f;                         // terms of f sorted by degree
    ==> 1+5y2+4z4+2x2y2z2+y4z3
    leadmonom(f);              // leading monomial
    ==> 1

    // Now we implement more "advanced" examples of rings:
    //
    // 1)   (K[y]_<y>)[x]
    //
    int n,m=2,3;
    ring A1 = 0,(x(1..n),y(1..m)),(dp(n),ds(m));
    poly f  = x(1)*x(2)^2+1+y(1)^10+x(1)*y(2)^5+y(3);
    leadmonom(f);
    ==> x(1)*x(2)^2
    leadmonom(1+y(1));         // unit
    ==> 1
    leadmonom(1+x(1));         // no unit
    ==> x(1)

    //
    // 2)  some ring in between (K[x]_<x>)[y] and K[x,y]_<x>
    //
    ring A2 = 0,(x(1..n),y(1..m)),(ds(n),dp(m));
    leadmonom(1+x(1));       // unit
    ==> 1
    leadmonom(1+x(1)*y(1));  // unit
    ==> 1
    leadmonom(1+y(1));       // no unit
    ==> y(1)

    //
    // 3)  K[x,y]_<x>
    //
    ring A4 = (0,y(1..m)),(x(1..n)),ds;
    leadmonom(1+y(1));       // in ground field
    ==> 1
    leadmonom(1+x(1)*y(1));  // unit
    ==> 1
    leadmonom(1+x(1));       // unit
    ==> 1
Note, that even if we implicitly compute over the localization of the polynomial ring, most computations are explicitly performed with polynomial data only. In particular, `1/(1-x);` does not return a power series expansion or a fraction but 0 (division with remainder in polynomial ring). See division for division with remainder in the localization and invunit for a procedure returning a truncated power series expansion of the inverse of a unit.

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