solve misses some solutions in a certain nonlinear system

[sagetrac-casamayou]

We search the critical points of the function x|-> (x2 + y2)^x

This function has four critical points : ±(0,1) and ±(1/e,0)

However the function solve can not find any of this.

More, solve returns (0,0) which is not a critical point since f is not differentiable at (0,0) !

  sage: var('x y')
  sage: f(x,y) = (x^2 + y^2)^x
  sage: solve([diff(f(x,y), x), diff(f(x,y), y)], x, y)
  [[x == 0, y == 0]] 

Upstream: Reported upstream. No feedback yet.

CC: @kcrisman @robert-marik @jasongrout

Component: calculus

Keywords: solve

Stopgaps: #12730

Issue created by migration from https://trac.sagemath.org/ticket/8862

[zimmermann6]

Changed author from Alexandre Casamayou to none

[burcin]

comment:2

What is the expected improvement in this ticket?

The solve() function in Sage is just a wrapper around maxima for now. In this case we just return the result from maxima.

There are two problems here:

• maxima's solve() cannot handle the given input, probably symbolic exponents defeat it
• maxima's solve() can return undefined points as output (solve() can return undefined points as "solutions" #2617)

If this ticket is about improving the capabilities of solve to handle the given input properly, this is an enhancement request. Do we know of any algorithm that will help with this?

Otherwise, this ticket is a duplicate of #2617.

Comments?

[zimmermann6]

comment:3

the issue here is not only that Sage returns undefined points (which indeed duplicates #2617) but
that it fails to find the following (trivial) solutions, which is a defect:

sage: sys=[diff(f(x,y), x), diff(f(x,y), y)]
sage: map(lambda s: s.subs(x=0,y=1),sys)
[0, 0]
sage: map(lambda s: s.subs(x=0,y=-1),sys)
[0, 0]
sage: map(lambda s: s.subs(x=1/e,y=0),sys)
[0, 0]
sage: map(lambda s: s.subs(x=-1/e,y=0),sys)
[0, 0]

For example Maple finds:

> f := (x,y) -> (x^2 + y^2)^x:                                         
> solve({diff(f(x,y), x), diff(f(x,y), y)}, {x, y}, Explicit=true);
                                          exp(1)               exp(1)
  {x = 0, y = 1}, {x = 0, y = -1}, {x = - ------, y = 0}, {x = ------, y = 0}
                                          exp(2)               exp(2)

When some solutions are lost, at least a warning should be issued.

[burcin]

comment:4

Replying to @zimmermann6:

the issue here is not only that Sage returns undefined points (which indeed duplicates #2617) but
that it fails to find the following (trivial) solutions, which is a defect:

> When some solutions are lost, at least a warning should be issued.

I don't think we can get that information out of maxima. Can someone more experienced in maxima comment on this? Or ask the maxima developers what they think about this problem?

Another option is to take this as an opportunity to start implementing some native solve() functionality in Sage. I have no idea how to (more or less algorithmically) find a solution to this system though. I'd appreciate any pointers.