# Michał Garmulewicz - bazy Groebnera

## 1761 days ago by HaskellCurry

# from sage.rings.polynomial.toy_buchberger import buchberger P1.<x,y,z> = PolynomialRing(ZZ, 3, order='lex') # possible orderings # * 'degrevlex' (default) – degree reverse lexicographic # * 'lex' – lexicographic # * 'deglex' – degree lexicographic f1 = x*y^2*z^2 + x*y - y*z # F = [x-y^2, y-z^3, z^2-1] F = [z^2-1, y - z^3, x - y^2] I = ideal(F) B = I.groebner_basis() print(I) print(B) print(f1.reduce(B)) print(f1.reduce(F))
 ```Ideal (z^2 - 1, y - z^3, x - y^2) of Multivariate Polynomial Ring in x, y, z over Integer Ring [x - 1, y - z, z^2 - 1] z z```
P2.<x,y,z> = PolynomialRing(ZZ, 3, order='deglex') f2 = x*y^2*z^2 + x*y - y*z # F = [x-y^2, y-z^3, z^2-1] F = [z^2-1, y - z^3, x - y^2] I_prime = ideal(F) B_prime = I_prime.groebner_basis() print(I_prime) print(B_prime) print(f2.reduce(B_prime)) print(f2.reduce(F))
 ```Ideal (z^2 - 1, -z^3 + y, -y^2 + x) of Multivariate Polynomial Ring in x, y, z over Integer Ring [z^2 - 1, x - 1, y - z] z x^2 + x*y - y*z```
P3.<x,y,z> = P1.change_ring(order='degrevlex') J = I.change_ring(P3) B2 = J.groebner_basis() print(B2) f1.reduce(B2)
 ```[z^2 - 1, x - 1, y - z] z```
P1.<x,y,z> = PolynomialRing(ZZ, 3, order='lex') F = [z^2-1, y - z^3, x - y^2] I = ideal(F) B = I.groebner_basis() print(I) print(B) print(ideal(B)) print(ideal(B)==I)
 ```Ideal (z^2 - 1, y - z^3, x - y^2) of Multivariate Polynomial Ring in x, y, z over Integer Ring [x - 1, y - z, z^2 - 1] Ideal (x - 1, y - z, z^2 - 1) of Multivariate Polynomial Ring in x, y, z over Integer Ring True```