|  |  7.7.2 Example of use of LETTERPLACE over Z 
Consider the following paradigmatic example:
 |  | LIB "freegb.lib";
ring r = integer,(x,y),Dp;
ring R = freeAlgebra(r,5); // length bound is 5
ideal I = 2*x, 3*y; 
I = twostd(I); 
print(matrix(I)); // pretty prints the generators
==> 3*y,2*x,y*x,x*y
 | 
 
As we can see, over 
 the ideal  has
a finite Groebner basis and indeed 
      
![$Z[x,y]/<2x,3y,xy>$](sing_1021.png) holds. 
Now, we analyze the same ideal in the ring with one more variable 
 : |  | LIB "freegb.lib";
ring r = integer,(x,y,z),Dp;
ring R = freeAlgebra(r,5,2); // length bound is 5
ideal I = 2*x, 3*y; 
I = twostd(I); 
print(matrix(I)); // pretty prints the generators
==> 3*y,2*x,y*x,x*y,y*z*x,x*z*y,y*z*z*x,x*z*z*y,y*z*z*z*x,x*z*z*z*y
 | 
 
Now we see, that this Groebner basis is potentially infinite and 
the following argument delivers a proof. Namely, 
 and 
 are present in the ideal for all  .
How can we do this? We wish to express  and 
 via the original generators: |  | LIB "freegb.lib";
ring r = integer,(x,y,z),Dp;
ring R = freeAlgebra(r,5,2); // length bound is 5, rank of the free bimodule is 2
ideal I = 2*x, 3*y; 
matrix T1 = lift(I, ideal(y*z*x,x*z*y)); 
print(T1);
==> -y*z*ncgen(1),-ncgen(1)*z*y,
==> ncgen(2)*z*x, x*z*ncgen(2)  
-y*z*I[1] + I[2]*z*x; // gives y*z*x
==> y*z*x
matrix T2 = lift(I, ideal(y*z^2*x,x*z^2*y)); 
print(T2);
==> -y*z*z*ncgen(1),-ncgen(1)*z*z*y,
==> ncgen(2)*z*z*x, x*z*z*ncgen(2)  
-y*z^2*I[1] + I[2]*z^2*x; // gives y*z^2*x
==> y*z*z*x
 | 
 
The columns of matrices, returned by lift, encode the presentation of new
elements in terms of generators. From this we conjecture, that in particular 
 holds for all  
and indeed, confirm it via a routine computation by hands.
 
Comparing computations over Q with computations over Z.
In the next example, we first compute over 
 and a bit later 
compare the result with computations over  . |  | LIB "freegb.lib"; // initialization of free algebras
ring r = 0,(z,y,x),Dp; // degree left lex ord on z>y>x
ring R = freeAlgebra(r,7); // length bound is 7
ideal I = y*x - 3*x*y - 3*z, z*x - 2*x*z +y, z*y-y*z-x;
option(redSB); option(redTail); // for minimal reduced GB
option(intStrategy); // avoid divisions by coefficients
ideal J = twostd(I); // compute a two-sided GB of I
J; // prints generators of J  
==> J[1]=4*x*y+3*z
==> J[2]=3*x*z-y
==> J[3]=4*y*x-3*z
==> J[4]=2*y*y-3*x*x
==> J[5]=2*y*z+x
==> J[6]=3*z*x+y
==> J[7]=2*z*y-x
==> J[8]=3*z*z-2*x*x
==> J[9]=4*x*x*x+x
LIB "fpadim.lib"; // load the library for K-dimensions
lpMonomialBasis(7,0,J); // compute all monomials 
==> _[1]=1
==> _[2]=z
==> _[3]=y
==> _[4]=x
==> _[5]=x*x
// of length up to 7 in Q<x,y,z>/J
 | 
 
As we see, we obtain a nice finite Groebner basis J.
Moreover, from the form of its leading monomials, we conjecture that 
 is finite dimensional  -vector space. 
We check it with lpMonomialBasisand obtain an affirmative answer. 
Now, for doing similar computations over 
 one needs to change
only the initialization of the ring, the rest stays the same |  | LIB "freegb.lib"; // initialization of free algebras
ring r = integer,(z,y,x),Dp; // Z and deg left lex ord on z>y>x
ring R = freeAlgebra(r,7); // length bound is 7
ideal I = y*x - 3*x*y - 3*z, z*x - 2*x*z +y, z*y-y*z-x;
option(redSB); option(redTail); // for minimal reduced GB
option(intStrategy); // avoid divisions by coefficients
ideal J = twostd(I); // compute a two-sided GB of I
J; // prints generators of J  
==> J[1]=12*x*y+9*z
==> J[2]=9*x*z-3*y
==> J[3]=y*x-3*x*y-3*z
==> J[4]=6*y*y-9*x*x
==> J[5]=6*y*z+3*x
==> J[6]=z*x-2*x*z+y
==> J[7]=z*y-y*z-x
==> J[8]=3*z*z+2*y*y-5*x*x
==> J[9]=6*x*x*x-3*y*z
==> J[10]=4*x*x*y+3*x*z
==> J[11]=3*x*x*z+3*x*y+3*z
==> J[12]=2*x*y*y+75*x*x*x+39*y*z+39*x
==> J[13]=3*x*y*z+3*y*y-3*x*x
==> J[14]=2*y*y*y+x*x*y+3*x*z
==> J[15]=2*x*x*x*x+y*y-x*x
==> J[16]=2*x*x*x*y+3*y*y*z+3*x*y+3*z
==> J[17]=x*x*y*z+x*y*y-x*x*x
==> J[18]=x*y*y*z-y*y*y+x*x*y
==> J[19]=x*x*x*x*x-y*y*y*z-x*y*y+x*x*x
==> J[20]=x*x*x*x*z+x*x*x*y+2*y*y*z+x*x*z+3*x*y+3*z
==> J[21]=x*y*y*y*z-y*y*y*y+x*x*x*x-y*y+x*x
==> J[22]=y*y*y*z*z-x*x*x*x*y
==> J[23]=x*y*y*y*y*z-y*y*y*y*y+x*x*y*y*y
==> J[24]=x*y*y*y*y*y*z-y*y*y*y*y*y+x*x*x*x*y*y+y*y*y*y+x*x*x*x+2*y*y-2*x*x
 | 
 
The output has plenty of elements in each degree (which is the same as length 
because of the degree ordering), what hints at potentially infinite Groebner basis. 
Indeed, one can show that for every 
 the ideal  contains an element 
with the leading monomial  . 
 |