# incOrbit -- the increasing map orbit of an element

## Synopsis

• Usage:
O = incOrbit(f,n)
O = incOrbit(F,n)
• Inputs:
• f, , 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 .