-1 in expression "is_positive".

[fmaltey]

I test :

y = I*I*x / x # so y is the expression -1
y._is_positive()  # the answer is True (and I dislike)
z = -x / x 
z._is_positive()   # the answer is False 
bool (z == y) # the answer is True

Apply attachment:trac_10064.patch .

[burcin]

Simpler example:

sage: (I*I)._is_positive()
True
sage: I*I
-1

The is_positive() test in pynac is at line 923 in numeric.cpp:

    case PYOBJECT:
      n = is_real() && (PyObject_Compare(v._pyobject, ZERO) > 0);
      if (PyErr_Occurred()) 
	py_error("is_positive");
      return n;

We use the python comparison. Unfortunately, this doesn't work as expected for number field elements:

sage: m = (I*I).pyobject()
sage: m.parent()
Number Field in I with defining polynomial x^2 + 1
sage: m > 0
True
sage: m
-1

[burcin]

attachment:trac_10064-compare_I.patch changes the quadratic number field element comparison code to return a more sensible result. Now we have:

sage: m = (I*I).pyobject()
sage: m > 0
False
sage: m < 0
True

This also fixes #10062.

[jgaski]

Here is a small bit of code that breaks (in 4.5.3) with trac_10064-compare_I.patch applied:

K.<a> = QuadraticField(5)
E = EllipticCurve(K,[-1, -1, 1/2*a + 1/2, -1/2*a - 1/2, 1/2*a + 1/2])
E.torsion_subgroup()

[jgaski]

Please see  http://tinypaste.com/3f27d for the error from the above code snippet.

[kcrisman]

Apparently related to #7160 and #6132, see  this sage-devel discussion.

[mhansen]

I've added a patch which verifies that is fixed by #7160.

Apply trac_10064.patch

[burcin]

Looks good to me.