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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
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As we can see, over
the ideal
has
a finite Groebner basis and indeed
and the later is naturally isomorphic to
as a
-algebra.
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); // 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
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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 by means of lift :
| 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
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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 the field of rationals
and
a bit later compare the result with computations over the ring of integers
.
| 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); // all monomials of length up to 7 in Q<x,y,z>/J
==> _[1]=1
==> _[2]=z
==> _[3]=y
==> _[4]=x
==> _[5]=x*x
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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 lpMonomialBasis and 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+6*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*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
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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
.
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