# reduce -- computes an equivariant normal form

## Synopsis

• Usage:
r = reduce(f,F)
• Inputs:
• f, , an element of a ring created by buildERing.
• F, a list, a list of polynomials from the same ring as f.
• Optional inputs:
• Completely => ..., default value false
• Outputs:
• r, , an equivariant normal form of f with respect to F.

## Description

Reduces f by the Inc-orbits of the set F until f is in a normal form. That is the lead monomial of f is a standard monomial, not divisible by any monomial in the orbit of any lead monomial of an element of F. If F is an equivariant Gröbner basis then r is 0 if and only if f is in the ideal generated by the orbits of F.

If the optional argument Completely is set to true then normal form r will contain only standard monomials. If F is an equivariant Gröbner basis, then the completely reduced normal form r is uniquely determined, otherwise there is no such guarantee.

 i1 : R = buildERing({symbol x}, {1}, QQ, 3); i2 : reduce(x_0^2 + x_0*x_2, {x_1}) 2 o2 = x 0 o2 : R

## Caveat

The output does not necessarily belong to the same ring as the input.

## Ways to use reduce :

• "reduce(RingElement,List)"

## For the programmer

The object reduce is .