Sage may ignore the imaginary part of variables not explicitly declared complex

[EmmanuelCharpentier]

Seen in this ask.sagemapt question, discussed in sage-devel :

Sage seems to assume that a not otherwise specified symbolic variable is real :

sage: var("z")
z
sage: real(z).simplify()
z
sage: imag(z).simplify()
0
sage: with assuming(z,"complex"): real(z).simplify()
realpart(z)
sage: with assuming(z,"complex"): imag(z).simplify()
imagpart(z)

As corollaries :

sage: var("a, b")
(a, b)
sage: (a*imag(z) + b*imag(z)).factor()
0
sage: (a*z + b*z).factor().imag_part()
imag_part(z)*real_part(a) + imag_part(z)*real_part(b) + imag_part(a)*real_part(z) + imag_part(b)*real_part(z)
sage: (a*z + b*z).factor().imag_part().factor()
0
sage: (a*z + b*z).factor().imag_part().simplify()
0

but :

sage: with assuming(z,"complex"): (a*z + b*z).factor().imag_part().factor()
(a + b)*imagpart(z)
sage: with assuming(z,"complex"): (a*z + b*z).factor().imag_part().simplify()
a*imagpart(z) + b*imagpart(z)

Maxima has the same problem : in Sage's maxima :

sage: %%maxima
....: domain;
....: imagpart(z);
....: imagpart(ratsimp(z));
....: imagpart(factor(z));
....: 
....: 
complex
0
0
0

but :

sage: %%maxima
....: declare(z,complex);
....: imagpart(z);
....: imagpart(ratsimp(z));
....: imagpart(factor(z));
....: 
....: 
done
'imagpart(z)
'imagpart(z)
'imagpart(z)

Filing this as blocker, because this problem may lead to Sage silently returning mathematical nonsense...

CC: @pjbruin

Component: symbolics

Issue created by migration from https://trac.sagemath.org/ticket/30793

[fchapoton]

comment:1

either blocker for the 9.3 milestone, or critical only, please

[EmmanuelCharpentier]

comment:2

Replying to @fchapoton:

either blocker for the 9.3 milestone, or critical only, please

Okay. Will do that also for #30688.

[vbraun]

comment:3

De facto this can't be a blocker if noone is enough invested to work on it. Probably should be fixed on the maxima side first, too.

[mkoeppe]

comment:4

Well, has anyone filed it as an upstream bug?

[DaveWitteMorris]

comment:5

This was reported upstream on #6862. The maxima developers deny that it is a bug, and I agree with them, because the maxima documentation makes it clear that merely setting domain:complex will not make individual variables default to being complex. If sage wants variables to be complex, then it should declare them that way.

However, I tried this a couple of months ago (just by making a minor modification to SR.symbol) and got lots of doctest failures. The two main problems were:

1. Maxima seems to give different (much more complicated) answers for some definite integrals when the integration variable is declared to be complex.
2. There were also a lot of failures in manifolds, so perhaps that code is assuming that the variables are real.

The problem has been around a long time, and I don't think this can be fixed in time for 9.3. We should try to do something for 9.4, but I think it will require serious work.

Maybe this ticket should be closed as a duplicate of #6862.

[mkoeppe]

comment:6

Thanks for the analysis! +1 for consolidating the two tickets, closing one of them, and moving the remaining one to milestone 9.4.

[nbruin]

comment:7

Replying to @DaveWitteMorris:

1. Maxima seems to give different (much more complicated) answers for some definite integrals when the integration variable is declared to be complex.

That's to be expected. Complex path integrals are generally path-dependent, so just specifying the end points of the path doesn't tell the whole story. Much of the integration machinery will normally be assuming real variables. This is one reason to not assume complex variables by default.

The main problem found on #6862 is the present incompatibility between pynac's and maxima's default. With this information, perhaps it would be worth seeing how bad inserting a default assume(<var>,"real") so that pynac at least agrees with maxima. If that leads to significantly less doctests, perhaps that's a preferable direction to resolve the mismatch.

[mkoeppe]

comment:8

Sage development has entered the release candidate phase for 9.3. Setting a new milestone for this ticket based on a cursory review.