sage: R.<a1,a2,b1,b2,b3>=QQ[]                                                                                                          
sage: eqs=[a1+b1, a2+a1*b1+b2, a2*b1+a1*b2+b3, a2*b2+a1*b3, a2*b3]                                                                     
sage: J=R.ideal(eqs)                                                                                                                   
sage: J.groebner_basis()                                                                                                               
[b1*b2^2 - 2*b2*b3, b2^3 - b3^2, b1*b2*b3 - b3^2, b2^2*b3, b1*b3^2, b2*b3^2, b3^3, a2^2 - b2^2 + b1*b3, a2*b1 - b1*b2 + b3, b1^2 - a2 - b2, a2*b2 - b1*b3, a2*b3, a1 + b1]
sage: J.dimension()                                                                                                                    
0
sage: J.vector_space_dimension()                                                                                                       
10
sage: binomial(5,2)                                                                                                                    
10
sage: pt=J.reduce(a2^3)                                                                                                                
sage: pt                                                                                                                               
b3^2
sage: R2.<x,y>=QQ[]                                                                                                                    
sage: (5*x) * (4*x+y) * (3*x+2*y) * (2*x+3*y) * (x+4*y) * (5*y)                                                                        
600*x^5*y + 3850*x^4*y^2 + 6725*x^3*y^3 + 3850*x^2*y^4 + 600*x*y^5
sage: (5*x) * (4*x+y) * (3*x+2*y) * (2*x+3*y) * (x+4*y) * (5*y) - (600*(x*y*(x+y)^4))                                                  
1450*x^4*y^2 + 3125*x^3*y^3 + 1450*x^2*y^4
sage: (5*x) * (4*x+y) * (3*x+2*y) * (2*x+3*y) * (x+4*y) * (5*y) - (600*(x*y*(x+y)^4)+1450*(x*y)^2*(x+y)^2)                             
225*x^3*y^3
sage: (5*x) * (4*x+y) * (3*x+2*y) * (2*x+3*y) * (x+4*y) * (5*y) - (600*(x*y*(x+y)^4)+1450*(x*y)^2*(x+y)^2+225*(x*y)^3)                 
0
sage: J.reduce(600*a2*a1^4+1450*a2^2*a1^2+225*a2^3)                                                                                    
2875*b3^2