incOrbit -- the increasing map orbit of an element

Synopsis

Usage:

O = incOrbit(f,n)

O = incOrbit(F,n)
Inputs:
- f, a ring element, an element of a ring created by buildERing
- F, a list, a list of ring elements
- n, an integer, the index bound
Optional inputs:
- Symmetrize => ..., default value false
Outputs:
- O, a list, a list of all elements in the orbit of f or F

Description

If F is an equivariant Gröbner basis for invariant ideal I with respect to a width order then this method produces a traditional Gröbner basis for the nth truncation of I.

If the optional argument Symmetrize is set to true, then the full S_n orbit is produced.

i1 : R = buildERing({symbol x}, {1}, QQ, 2);

i2 : O = incOrbit(x_0^2, 4)

       2   2   2   2
o2 = {x , x , x , x }
       0   1   2   3

o2 : List

i3 : P = incOrbit(x_0 + x_1^2, 3, Symmetrize=>true)

       2             2   2             2   2             2
o3 = {x  + x , x  + x , x  + x , x  + x , x  + x , x  + x }
       1    0   1    0   2    0   2    0   2    1   2    1

o3 : List

Caveat

The output is not necessarily in the same ring as the input. The width bound of the ring of the output will always be n.

Ways to use `incOrbit` :

"incOrbit(List,ZZ)"
"incOrbit(RingElement,ZZ)"

For the programmer