
D.8.2.11 sortandmap
Procedure from library presolve.lib (see presolve_lib).
 Usage:
 sortandmap(id [,n1,p1,n2,p2...,o1,m1,o2,m2...]);
id=poly/ideal/vector/module,
p1,p2,...= polynomials (product of variables),
n1,n2,...= integers,
o1,o2,...= strings,
m1,m2,...= integers
(default: p1=product of all vars, n1=0, o1="dp",m1=0)
the last pi (containing the remaining vars) may be omitted
 Return:
 a ring R, in which a poly/ideal/vector/module IMAG is stored:
 the ring R differs from the active basering only in the choice
of monomial ordering and in the sorting of the variables.
 IMAG is the image (under imap) of the input ideal/module id
The new monomial ordering and sorting of vars is as follows:
  each block of vars occuring in pi is sorted w.r.t. its complexity in id,
 ni controls the sorting in ith block (= vars occuring in pi):
ni=0 (resp. ni!=0) means that least complex (resp. most complex) vars come
first
 oi and mi define the monomial ordering of the ith block:
if mi =0, oi=ordstr(ith block)
if mi!=0, the ordering of the ith block itself is a blockordering,
each subblock having ordstr=oi, such that vars of same complexity are
in one block
 Note that only simple ordstrings oi are allowed: "lp","dp","Dp",
"ls","ds","Ds".
 Note:
 We define a variable x to be more complex than y (with respect to id)
if val(x) > val(y) lexicographically, where val(x) denotes the
valuation vector of x:
consider id as list of polynomials in x with coefficients in the
remaining variables. Then:
val(x) = (maximal occuring power of x, # of all monomials in leading
coefficient, # of all monomials in coefficient of next smaller power
of x,...).
Example:
 LIB "presolve.lib";
ring s = 32003,(x,y,z),dp;
ideal i=x3+y2,xz+z2;
def R_r=sortandmap(i);
==>
==> // 'sortandmap' created a ring, in which an object IMAG is stored.
==> // To access the object, type (if the name R was assigned to the return v\
alue):
==> setring R; IMAG;
show(R_r);
==> // ring: (ZZ/32003),(x,y,z),(dp(3),C);
==> // minpoly = 0
==> // objects belonging to this ring:
==> // IMAG [0] ideal, 2 generator(s)
setring R_r; IMAG;
==> IMAG[1]=x3+y2
==> IMAG[2]=xz+z2
kill R_r; setring s;
def R_r=sortandmap(i,1,xy,0,z,0,"ds",0,"lp",0);
==>
==> // 'sortandmap' created a ring, in which an object IMAG is stored.
==> // To access the object, type (if the name R was assigned to the return v\
alue):
==> setring R; IMAG;
show(R_r);
==> // ring: (ZZ/32003),(x,y,z),(ds(2),lp(1),C);
==> // minpoly = 0
==> // objects belonging to this ring:
==> // IMAG [0] ideal, 2 generator(s)
setring R_r; IMAG;
==> IMAG[1]=y2+x3
==> IMAG[2]=z2+xz
kill R_r;

