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

The object incOrbit is a method function with options.