Bug in solve due to a bug in symbolic_expression_from_maxima_string

[bgrenet]

A bug in sage.calculus.calculus.symbolic_expression_from_maxima_string implies bugs in solve and roots for symbolic expressions.

Symptoms

The method solve for symbolic expressions is buggy (the list of multiplicities has size 2 instead of 4), as well as roots as a (serious!) consequence:

sage: w = x^4 - (1+3*i)*x^3 - (2-4*i)*x^2 + (6-2*i)*x - 4 - 4*i
sage: w.solve(x,multiplicities=True)
([x == -1/2*sqrt(2*I) + 3/2*I - 1/2, x == 1/2*sqrt(2*I) + 3/2*I - 1/2, x == (-I + 1), x == (I + 1)],
 [1, 1])
sage: w.roots() # should be 4 roots
[(-1/2*sqrt(2*I) + 3/2*I - 1/2, 1), (1/2*sqrt(2*I) + 3/2*I - 1/2, 1)]

Diagnosis

1. The behavior of roots is easily explained by the behavior of solve, since roots assume that the length of the list of solutions is the same as the length of the list of multiplicities. This should clearly be the case so the bug is not in roots.

2. Given the parameter in the example, solve calls Maxima and parses the result. The multiplicities are obtained by invoking P.get('multiplicities'). This is the right invocation to Maxima, no bug there.

3. To parse the solutions returned by Maxima, solve calls sage.symbolic.relation.string_to_list_of_solutions, which itself calls sage.calculus.symbolic_expression_from_maxima_string. The bug occurs in this last function: Indeed, while there is no apparent reason for this, invoking this function changes the variable multiplicities of Maxima. Here is an example:

   sage: w = x^4 - (1+3*i)*x^3 - (2-4*i)*x^2 + (6-2*i)*x - 4 - 4*i
   sage: m = (w == 0)._maxima_()
   sage: P = m.parent()
   sage: s = m.solve(x).str()
   sage: s # the list of solutions returned by Maxima
   '[_SAGE_VAR_x=-(sqrt(2*%i)-3*%i+1)/2,_SAGE_VAR_x=(sqrt(2*%i)+3*%i-1)/2,_SAGE_VAR_x=1-%i,_SAGE_VAR_x=%i+1]'
   sage: P.get('multiplicities') # correct!
   '[1,1,1,1]'
   sage: l = sage.calculus.calculus.symbolic_expression_from_maxima_string(s)
   sage: l
   [x == -1/2*sqrt(2*I) + 3/2*I - 1/2,
    x == 1/2*sqrt(2*I) + 3/2*I - 1/2,
    x == (-I + 1),
    x == (I + 1)]
   sage: P.get('multiplicities') # WTF???
   '[1,1]'

Component: **symbolics**

Keywords: **maxima, solve**

Author: **Bruno Grenet**

Branch/Commit: **[`eb0ad68`](https://github.com/sagemath/sagetrac-mirror/commit/eb0ad68ae007807828945e13e58e9865c67adf77)**

Reviewer: **Vincent Delecroix**

_Issue created by migration from https://trac.sagemath.org/ticket/20755_

[nbruin]

comment:2

Well, multiplicities is a system variable. See http://maxima.sourceforge.net/docs/manual/maxima_20.html. Who knows when it gets updated? In particular, the reconstruction of symbolic expressions from strings might be calling into maxima again.

The quick fix is to get the correct value of multiplicities out as soon as we can, before conversion back.

In the longer run, it would be preferable to do the conversion while avoiding strings altogether. See sr_sum and sr_limit etc. for examples how one can use sr_to_max and max_to_sr to convert.

[fchapoton]

comment:3

seems to work in 8.9.b1

one needs to check and add a doctest

[bgrenet]

Commit: eb0ad68

[bgrenet]

comment:4

The bug seems indeed fixed. I've add a doctest.

---

New commits:

eb0ad68

20755: Add doctest

[bgrenet]

Author: Bruno Grenet

[bgrenet]

Branch: u/bruno/20755_bug_in_solve

[videlec]

Reviewer: Vincent Delecroix