You can check what ring each ideal is in like `I5.ring()`

. In this case I get `Symbolic Ring`

. Presumably it converts to this because Sage treats `sqrt(2)`

as an exact symbolic expression and does not presume to convert it to something less precise even if you use it in an expression with variables like `x`

and `y`

that are in your ring on `RR`

. (In this case I feel like it *should* be able to do just what you clearly mean, but I'm not expert-enough on this to know if there's really a better way to do this unambiguously). One way (of several) to fix this would be to use the `.ideal`

method on your ring like `R.ideal([(x^2+y^2-1),(x*y),(y^3-y)])`

. This ensures that each of the generators can be converted to an element of your ring. I get:

```
sage: I5 = R.ideal([x-1/sqrt(2),y-1/sqrt(2)])
sage: I5
Ideal (x - 0.707106781186548, y - 0.707106781186548) of Multivariate Polynomial Ring in x, y over Real Field with 53 bits of precision
```

And so on.

Unfortunately when I try to take the intersection I get an exception:

```
TypeError: Cannot call Singular function 'intersect' with ring parameter of type '<class 'sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict_domain_with_category'>'
```

If I try things like:

```
sage: R.<x,y>=PolynomialRing(RR,order='lex',implementation='singular')
```

I get:

```
NotImplementedError: polynomials over Real Field with 53 bits of precision are not supported in Singular
```

which is strange because the Singular docs seem to say so.

I'm not an algebraicist or even a mathematician at all so I'm in over my head, but maybe you can try using a number field ( $\mathbb{Q}$ extended by $\sqrt 2$)? It certainly gives *an* answer which AFAICT should be just as applicable if $ x $ and $ y $ are real. I'm just not sure if it's the one you're looking for. Perhaps it would help if you gave more details about the specific problem you're trying to solve:

```
sage: F.<sqrt2> = NumberField(x^2 - 1/2)
sage: F
Number Field in sqrt2 with defining polynomial x^2 - 1/2
sage: R.<x,y> = PolynomialRing(F, order='lex')
sage: R
Multivariate Polynomial Ring in x, y over Number Field in sqrt2 with defining polynomial x^2 - 1/2
sage: I = R.ideal([(x^2+y^2-1),(x*y),(y^3-y)])
sage: I5 = R.ideal([x-1/sqrt2,y-1/sqrt2])
sage: I6 = R.ideal([x+1/sqrt2,y-1/sqrt2])
sage: I7 = R.ideal([x+1/sqrt2,y+1/sqrt2])
sage: I8 = R.ideal([x-1/sqrt2,y+1/sqrt2])
sage: I.intersection(I5, I6, I7, I8)
Ideal (2*y^5 - 3*y^3 + y, 2*x*y^3 - x*y, x^2 + y^2 - 1) of Multivariate Polynomial Ring in x, y over Number Field in sqrt2 with defining polynomial x^2 - 2
```

Hopefully someone with more expertise in this problem area can give a better answer.

I fixed the block quoting. Not sure why you were having trouble.