# forceGB(...,SyzygyMatrix=>...) -- specify the syzygy matrix

## Synopsis

• Usage:
forceGB(f,SyzygyMatrix=>z,...)
• Inputs:
• z,
• Consequences:
• A request for the syzygy matrix of f will return z

## Description

In the following example, the only computation being performed when asked to compute the kernel or syz of f is the minimal generator matrix of z.
 i1 : gbTrace = 3 o1 = 3 i2 : R = ZZ[x,y,z]; -- registering polynomial ring 3 at 0x7fe89b36d600 i3 : f = matrix{{x^2-3, y^3-1, z^4-2}}; 1 3 o3 : Matrix R <--- R i4 : z = koszul(2,f) o4 = {2} | -y3+1 -z4+2 0 | {3} | x2-3 0 -z4+2 | {4} | 0 x2-3 y3-1 | 3 3 o4 : Matrix R <--- R i5 : g = forceGB(f, SyzygyMatrix=>z); i6 : syz g -- no extra computation o6 = {2} | -y3+1 -z4+2 0 | {3} | x2-3 0 -z4+2 | {4} | 0 x2-3 y3-1 | 3 3 o6 : Matrix R <--- R i7 : syz f -- registering gb 0 at 0x7fe89b3b2e00 -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb elements = 3 -- number of monomials = 9 -- #reduction steps = 6 -- #spairs done = 6 -- ncalls = 0 -- nloop = 0 -- nsaved = 0 -- o7 = {2} | -y3+1 -z4+2 0 | {3} | x2-3 0 -z4+2 | {4} | 0 x2-3 y3-1 | 3 3 o7 : Matrix R <--- R i8 : kernel f o8 = image {2} | -y3+1 -z4+2 0 | {3} | x2-3 0 -z4+2 | {4} | 0 x2-3 y3-1 | 3 o8 : R-module, submodule of R
If you know that the columns of z already form a set of minimal generators, then one may use forceGB once again.

## Further information

• Default value: null
• Function: forceGB -- declare that the columns of a matrix are a Gröbner basis
• Option key: SyzygyMatrix -- an optional argument

## Caveat

If the columns of z do not generate the syzygy module of f, nonsensical answers may result