Singular Manual: Examples of use of Letterplace over Z

| ### 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 309#309 the ideal 310#310 has a finite Groebner basis and indeed 311#311 312#312 313#313 and the later is naturally isomorphic to 314#314 as a 30#30-algebra. Now, we analyze the same ideal in the ring with one more variable 42#42:
    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
Now we see, that this Groebner basis is potentially infinite and the following argument delivers a proof. Namely, 315#315 and 316#316 are present in the ideal for all 317#317. How can we do this? We wish to express 318#318 and 319#319 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
The columns of matrices, returned by `lift`, encode the presentation of new elements in terms of generators. From this we conjecture, that in particular 320#320 holds for all 317#317 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 3#3 and a bit later compare the result with computations over the ring of integers 30#30.
    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
As we see, we obtain a nice finite Groebner basis `J`. Moreover, from the form of its leading monomials, we conjecture that 321#321 is finite dimensional 3#3-vector space. We check it with `lpMonomialBasis` and obtain an affirmative answer. Now, for doing similar computations over 30#30 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
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 322#322 the ideal 278#278 contains an element with the leading monomial 323#323.

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