# egbToric -- computes the kernel of an equivariant monomial map

## Synopsis

• Usage:
G = egbToric m
• Inputs:
• m, , an equivariant monomial map between rings with symmetric action
• Optional inputs:
• OutFile => ..., default value null, where to send messages
• Outputs:
• G, a list, an equivariant Gröbner basis for the kernel of the map

## Description

m should be a monomial map between rings created by buildERing. Such a map can be constructed with buildEMonomialMap but this is not required.

For a map to ring R from ring S, the algorithm infers the entire equivariant map from where m sends the variable orbit generators of S. In particular for each orbit of variables of the form x_{(i_1,...,i_k)}, the image of x_{(0,...,k-1)} is used.

egbToric uses an incremental strategy, computing Gröbner bases for truncations using FourTiTwo. Because of FourTiTwo's efficiency, this strategy tends to be much faster than general equivariant Gröbner basis algorithms such as egb.

In the following example we compute an equivariant Gröbner basis for the vanishing equations of the second Veronese of P^n, i.e. the variety of n x n rank 1 symmetric matrices.

 i1 : R = buildERing({symbol x}, {1}, QQ, 2); i2 : S = buildERing({symbol y}, {2}, QQ, 2); i3 : m = buildEMonomialMap(R,S,{x_0*x_1}) 2 2 o3 = map (R, S, {x , x x , x x , x }) 1 1 0 1 0 0 o3 : RingMap R <--- S i4 : G = egbToric(m, OutFile=>stdio) 3 -- used .00118096 seconds -- used .000119554 seconds (9, 9) new stuff found 4 -- used .00240077 seconds -- used .000974549 seconds (16, 26) new stuff found 5 -- used .00473977 seconds -- used .00286254 seconds (25, 60) 6 -- used .0107124 seconds -- used .00677128 seconds (36, 120) 7 -- used .0263283 seconds -- used .0405163 seconds (49, 217) 2 o4 = {- y + y , - y y + y , - y y + y y , - y y + 1,0 0,1 1,1 0,0 1,0 2,1 0,0 2,0 1,0 2,1 1,0 ------------------------------------------------------------------------ y y , - y y + y y , - y y + y y , - y y + 2,0 1,1 2,2 1,0 2,1 2,0 3,2 1,0 3,0 2,1 3,2 1,0 ------------------------------------------------------------------------ y y } 3,1 2,0 o4 : List

## Caveat

It is not checked if m is equivariant. Only the images of the orbit generators of the source ring are examined and the rest of the map ignored.