Singular

D.8.2.13 valvars

Usage:

valvars(id[,n1,p1,n2,p2,...]);
id=poly/ideal/vector/module,
p1,p2,...= polynomials (product of vars),
n1,n2,...= integers,

ni controls the ordering of vars occurring in pi: ni=0 (resp. ni!=0) means that less (resp. more) complex vars come first (default: p1=product of all vars, n1=0),
the last pi (containing the remaining vars) may be omitted

Compute:

valuation (complexity) of variables with respect to id.
ni controls the ordering of vars occurring in pi:
ni=0 (resp. ni!=0) means that less (resp. more) complex vars come first.

Return:

list with 3 entries:

[1]: intvec, say v, describing the permutation such that the permuted ring variables are ordered with respect to their complexity in id [2]: list of intvecs, i-th intvec, say v(i) describing permutation of vars in a(i) such that v=v(1),v(2),... [3]: list of ideals and intmat's, say a(i) and M(i), where a(i): factors of pi, M(i): valuation matrix of a(i), such that the j-th column of M(i) is the valuation vector of j-th generator of a(i)

Note:

Use sortvars in order to actually sort the variables! 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 occurring power of x, # of all monomials in leading coefficient, # of all monomials in coefficient of next smaller power of x,...).

LIB "presolve.lib";
ring s=0,(x,y,z,a,b),dp;
ideal i=ax2+ay3-b2x,abz+by2;
valvars (i,0,xyz);
==> [1]:
==>    1,2,3,4,5
==> [2]:
==>    [1]:
==>       3,1,2
==>    [2]:
==>       1,2
==> [3]:
==>    [1]:
==>       _[1]=x
==>       _[2]=y
==>       _[3]=z
==>    [2]:
==>       2,3,1,
==>       1,1,1,
==>       1,1,0 
==>    [3]:
==>       _[1]=a
==>       _[2]=b
==>    [4]:
==>       1,2,
==>       3,1,
==>       0,2