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The first example you cite is about points on a curve. Your variety is the intersection of two curves ($y = x^2$ and $y = x - 1$), which is a finite set of points (and not a curve), so you cannot use the same technique. What you did later is correct:
sage: R.<x,y> = PolynomialRing(QQ)
sage: I = R.ideal(y-x^2,y-x+1)
sage: I.variety()
[]
This tells you that the set of points in the variety which are defined over $\mathbb{Q}$ (i.e. which have coordinates in $\mathbb{Q}$) is empty. You can also try to find (numerically) points defined over $\mathbb{R}$:
sage: I.variety(RR)
[]
Again it is an empty set, because the two curves don't intersect when drawn in $\mathbb{R}^2$. You can try to find (numerically) points defined over $\mathbb{C}$:
sage: I.variety(CC)
[{y: -0.500000000000000 - 0.866025403784439*I, x: 0.500000000000000 - 0.866025403784439*I},
{y: -0.500000000000000 + 0.866025403784439*I, x: 0.500000000000000 + 0.866025403784439*I}]
Here we find two points in $\mathbb{C}^2$. If you are looking for exact answers, you can also try to find points defined over $\bar{\mathbb{Q}}$ (the algebraic closure of $\mathbb{Q}$):
sage: I.variety(QQbar)
[{y: -0.50000000000000000? - 0.866025403784439?*I, x: 0.50000000000000000? - 0.866025403784439?*I},
{y: -0.50000000000000000? + 0.866025403784439?*I, x: 0.50000000000000000? + 0.866025403784439?*I}]
The result looks numerical, but Sage knows exactly what they are, and you can do exact arithmetic with these points. In this case you can also express them as radicals:
sage: [(P[x].radical_expression(), P[y].radical_expression()) for P in I.variety(QQbar)]
[(-1/2*I*sqrt(3) + 1/2, -1/2*I*sqrt(3) - 1/2),
(1/2*I*sqrt(3) + 1/2, 1/2*I*sqrt(3) - 1/2)]
This is similar to what you would get if you asked Sage to solve the equations symbolically:
sage: solve(map(SR, I.gens()), map(SR, R.gens()))
[[x == -1/2*I*sqrt(3) + 1/2, y == -1/2*I*sqrt(3) - 1/2], [x == 1/2*I*sqrt(3) + 1/2, y == 1/2*I*sqrt(3) - 1/2]]
| | 2 | No.2 Revision |
The first example you cite is about points on a curve. Your variety is the intersection of two curves ($y = x^2$ and $y = x - 1$), which is a finite set of points (and not a curve), so you cannot use the same technique. What technique.
I assume by "affine variety" you mean affine variety defined over $\mathbb{Q}$, and by "rational points" you mean points defined over $\mathbb{Q}$. In that case, what you did later is correct:
sage: R.<x,y> = PolynomialRing(QQ)
sage: I = R.ideal(y-x^2,y-x+1)
sage: I.variety()
[]
This tells you that the set of points in the variety which are defined over $\mathbb{Q}$ (i.e. which have coordinates in $\mathbb{Q}$) is empty. You can also try to find (numerically) points defined over $\mathbb{R}$:
sage: I.variety(RR)
[]
Again it is an empty set, because the two curves don't intersect when drawn in $\mathbb{R}^2$. You can try to find (numerically) points defined over $\mathbb{C}$:
sage: I.variety(CC)
[{y: -0.500000000000000 - 0.866025403784439*I, x: 0.500000000000000 - 0.866025403784439*I},
{y: -0.500000000000000 + 0.866025403784439*I, x: 0.500000000000000 + 0.866025403784439*I}]
Here we find two points in $\mathbb{C}^2$. If you are looking for exact answers, you can also try to find points defined over $\bar{\mathbb{Q}}$ (the algebraic closure of $\mathbb{Q}$):
sage: I.variety(QQbar)
[{y: -0.50000000000000000? - 0.866025403784439?*I, x: 0.50000000000000000? - 0.866025403784439?*I},
{y: -0.50000000000000000? + 0.866025403784439?*I, x: 0.50000000000000000? + 0.866025403784439?*I}]
The result looks numerical, but Sage knows exactly what they these points are, and you can do exact arithmetic with these points. them. In this case you can also express them as radicals:
sage: [(P[x].radical_expression(), P[y].radical_expression()) for P in I.variety(QQbar)]
[(-1/2*I*sqrt(3) + 1/2, -1/2*I*sqrt(3) - 1/2),
(1/2*I*sqrt(3) + 1/2, 1/2*I*sqrt(3) - 1/2)]
This is similar to what you would get if you asked Sage to solve the equations symbolically:
sage: solve(map(SR, I.gens()), map(SR, R.gens()))
[[x == -1/2*I*sqrt(3) + 1/2, y == -1/2*I*sqrt(3) - 1/2], [x == 1/2*I*sqrt(3) + 1/2, y == 1/2*I*sqrt(3) - 1/2]]
