Mea Culpa - Cleaning up mistakes from last time
=================================
###
R.<x,y> = ZZ[]

S.<a,b> = R.quotient((x^2 + y^2, 17))

S
### The following computation shows that the result of a computation is not necessarily reduced: 
(a+b)^17

### This construction actually has different behaviour: 
R.<x,y> = Zmod(17)[]

S.<a,b> = R.quotient((x^2 + y^2,))

S

(a+b)^17



New content
=========================================================


###Elimination ideals
R.<x,y,t,s,z> = PolynomialRing(QQ,5)

I = R * [x-t,y-t^2,z-t^3,s-x+y^3]

J = I.elimination_ideal([t,s]); J




####An example from class

#Let’s consider the map F^2 → F^33 given by (t, u) |-> (t(u^2 − t^22), u, u^2 − t^2). What’s a minimal set of equations describing its image?


R.<t, u, x, y, z> = PolynomialRing(QQ, order=’lex’)
I = R.ideal([x - t*(u^2 - t^2), y - u, z - (u^2 - t^2)])
I.groebner_basis()

# As the last polynomial does not involve u,t, the ideal of polynomials vanishing on the image of \bar F^2 is generated by x^2 - y^2*z^2 + z^3

# Alternative solution:
I.elimination_ideal([t, u])



### (Affine) Algebraic schemes in sage
A.<x, y> = AffineSpace(2, QQ)

X = A.subscheme([x^2-y])

Y = A.subscheme([y])

X.union(Y)
Z= X.intersection(Y)
Z.defining_polynomials() 
 Z.defining_ideal()
Z.coordinate_ring() 

#Other schemes
X = A.subscheme([x^2-x])
Y = A.subscheme([x^2-2*x])
X.irreducible_components() 

C=A.subscheme([x])  
C==A



##
X = A.subscheme([x^2-x]) 
Y = A.subscheme([x^2-2*x]) 
Z= X.intersection(Y)

######An example application of Gröbner bases:

# This example shows how we can use Groebner bases over Z to find the primes modulo which a system of equations has a solution, when the system has no solutions over the rationals.
#We first form a certain ideal I in Z[x,y,z], and note that the Groebner basis of I over Q contains 1, so there are no solutions over Q or an algebraic closure of it (this is not surprising as there are 4 equations in 3 unknowns).

P.<x,y,z> = PolynomialRing(ZZ,order='lex')

I = ideal(-y^2 - 3*y + z^2 + 3, -2*y*z + z^2 + 2*z + 1, x*z + y*z + z^2, -3*x*y + 2*y*z + 6*z^2)

I.change_ring(P.change_ring(QQ)).groebner_basis()

#Thus, there are no solutions over Q or an algebraic extension of it. 

I.groebner_basis()

###Thus we will get a non-trivial Gröbner basis of I mod p for each p dividing 164878

factor(164878)
2 * 7 * 11777

I.change_ring(P.change_ring( GF(2) )).groebner_basis()


I.change_ring(P.change_ring( GF(7) )).groebner_basis()


I.change_ring(P.change_ring( GF(11777 ))).groebner_basis()