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Define the affine variety (a) $X = V (y − x^2, y − x + 1)$. (b) Find all the rational points on X.

What I got in the examples is that we can use the code sage: x,y,z = PolynomialRing(GF(5), 3, 'xyz').gens() sage: C = Curve(y^2z^7 - x^9 - xz^8); C sage: C.rational_points()

To get rational points over Finite Field of size 5. To calculate over rational we can replace $GF(5)$ by QQ but to get a finite result we have to have the intersection.

Define the affine variety (a) $X = V (y − x^2, y − x + 1)$. (b) Find all the rational points on X.

What I got in the examples is that we can use the code sage: x,y,z = PolynomialRing(GF(5), 3, 'xyz').gens() sage: C = Curve(y^2z^7 - x^9 - xz^8); C sage: C.rational_points()

To get rational points over Finite Field of size 5. To calculate over rational we can replace $GF(5)$ by QQ but to get a finite result we have to have the intersection.

Define the affine variety (a) $X = V (y − x^2, y − x + 1)$. (b) Find all the rational points on X.

What I got in the examples is that we can use the code code

sage: x,y,z = PolynomialRing(GF(5), 3, 'xyz').gens() 'xyz').gens()

sage: C = Curve(y^2z^7 - x^9 - xz^8); C C

sage: C.rational_points()

To get rational points over Finite Field of size 5. To calculate over rational we can replace $GF(5)$ by QQ but to get a finite result we have to have the intersection.

Later I also used: used:

sage: R.<x,y> = PolynomialRing(QQ) PolynomialRing(QQ)

sage: R R

sage: I = R.ideal(y-x^2,y-x+1) R.ideal(y-x^2,y-x+1)

sage: I.variety()

But didn't get my result.