# Singular Book 1.8.4 -- elimination of variables

There are several methods to eliminate variables in Macaulay2.
 i1 : A = QQ[t,x,y,z]; i2 : I = ideal"t2+x2+y2+z2,t2+2x2-xy-z2,t+y3-z3"; o2 : Ideal of A i3 : eliminate(I,t) 2 2 2 6 3 3 6 2 2 o3 = ideal (x - x*y - y - 2z , y - 2y z + z + x*y + 2y + 3z ) o3 : Ideal of A

Alternatively, one may do it by hand: the elements of the Groebner basis under an elimination order not involving t generate the elimination ideal.

 i4 : A1 = QQ[t,x,y,z,MonomialOrder=>{1,3}]; i5 : I = substitute(I,A1); o5 : Ideal of A1 i6 : transpose gens gb I o6 = {-2} | x2-xy-y2-2z2 | {-6} | y6-2y3z3+z6+xy+2y2+3z2 | {-3} | t+y3-z3 | 3 1 o6 : Matrix A1 <--- A1

Here is another elimination ideal. Weights not given are assumed to be zero.

 i7 : A2 = QQ[t,x,y,z,MonomialOrder=>Weights=>{1}]; i8 : I = substitute(I,A2); o8 : Ideal of A2 i9 : transpose gens gb I o9 = {-2} | x2-xy-y2-2z2 | {-6} | y6-2y3z3+z6+xy+2y2+3z2 | {-3} | t+y3-z3 | 3 1 o9 : Matrix A2 <--- A2

The same order as the previous one:

 i10 : A3 = QQ[t,x,y,z,MonomialOrder=>Eliminate 1]; i11 : I = substitute(I,A3); o11 : Ideal of A3 i12 : transpose gens gb I o12 = {-2} | x2-xy-y2-2z2 | {-6} | y6-2y3z3+z6+xy+2y2+3z2 | {-3} | t+y3-z3 | 3 1 o12 : Matrix A3 <--- A3