nc_setup({k,n,NN,KK},{NN*n-n*NN=NN,KK*k-k*KK=KK},left);
p1 := (n-k+1)*NN - (n+1);
p2 := (k+1)*KK -(n-k);
l_g:=nc_groebner ({p1,p2});
nc_preduce(p1+p2,l_g);
nc_divide (k*p1+p2,p1);
nc_divide (k*p1+p2,2*p1);
nc_divide (2*k*k*p1 + k*p1 + p2,2*p1);
nc_factorize (p1*p2);
nc_setup({k,n,NN,KK},{NN*n-n*NN=NN,KK*k-k*KK=KK},right);
nc_factorize (p1*p2);
% applications to shift operators
nc_setup({n,NN},{NN*n-n*NN=1},left);
n*NN;
nc_factorize(ws);
nc_setup({n,NN},{NN*n-n*NN=1},right);
n*NN;
nc_factorize(ws);
nc_setup({NN,n},{NN*n-n*NN=1},right);
n*NN;
nc_factorize(ws);
nc_setup({NN,n},{NN*n-n*NN=1},left);
n*NN;
nc_factorize(ws);
% Applications to partial differential equations
nc_setup({x,Dx},{Dx*x-x*Dx=1});
p:= 2*Dx^2 + x* Dx^3 + 3*x*Dx + x^2*Dx^2 + 14 + 7*x*Dx;
nc_factorize p;
right_factor(p,1); % no factor of degr 1
right_factor(p,2);
left_factor(p,2);
nc_setup({x,Dx},{Dx*x-x*Dx=1});
q := x**2*dx**2 + 2*x**2*dx + x*dx**3 + 2*x*dx**2
+ 8*x*dx + 16*x + 2*dx**2 + 4*dx$
nc_factorize q;
right_factor(q,1);
right_factor(q,1,{x}); % no such right factor
right_factor(q,1,{dx});
% looking for factor with degree bound for an individual variable
q := x**6*dx + x**5*dx**2 + 12*x**5 + 10*x**4*dx + 20*x**3
+ x**2*dx**3 - x**2*dx**2 + x*dx**4 - x*dx**3 + 8*x*dx**2
- 8*x*dx + 2*dx**3 - 2*dx**2$
right_factor(q,dx);
right_factor(q,dx^2);
% some coefficient sports
nc_setup({NN,n},{NN*n-n*NN=1},left);
q:=(n*nn)^2;
nc_factorize q;
nc_preduce(q,{c1+c2*n + c3*nn + c4*n*nn});
nc_divide(q,n);
nc_cleanup;
end;