precalculus algebra: find the solution set of a complicated system of conditions (complex numbers)

Leave $ x, and in mathbb {C ^ *} $ have the following properties:

$ | x | ^ 2 + | and | ^ 2 = 1 $, $ x ^ n + y ^ m> 0 $ ($ x, y $ they are complex numbers but this sum is strictly positive real) where $ gcd (n, m) = 1 $

Leave $ Delta theta in mathbb {R} $, $ Delta varphi in mathbb {R} $ Y $ Delta t in mathbb {R} $ be such that

$ e ^ {i ( Delta theta + Delta t / n)} = 1 $ (*)

$ e ^ {i ( Delta varphi + Delta t / m)} = 1 $ (**)

Y

$ (xe ^ {i Delta theta}) ^ n + (ye ^ {i Delta varphi}) ^ m> 0 $ (***)

Is it true that, under these conditions, the only solutions are $ Delta t = 2k pi $, $ Delta theta equiv- frac { Delta t} {n} pmod {2 pi m} $ Y $ Delta varphi equiv – frac { Delta t} {m} pmod {2 pi n} $ for $ k en mathbb {Z} $? I know these are solutions, but are they the only ones?


I've been trying to get this result in vain. I'm not sure of the condition $ | x | ^ 2 + | and | ^ 2 = 1 $ It is useful.

I know that the conditions (*) and (**) imply:

$ Delta theta + frac { Delta t} {n} equiv0 pmod {2 pi} $

$ Delta varphi + frac { Delta t} {m} equiv0 pmod {2 pi} $

which is close to what I want but not exactly.

It is much less clear to me what I can get from (***) knowing that $ x ^ n + y ^ m> 0 $. A geometric interpretation helps if $ x $ Y $ and $ They are conjugated, but otherwise it is more difficult. As $ x $ Y $ and $ are complex numbers, the algebra quickly becomes complicated … I think it might be possible to obtain from this a relationship between $ Delta theta $ Y $ Delta varphi $, and maybe conclude that at least $ Delta t $ is a multiple of $ 2 pi $, but I have not managed to do it in my scribbles.

Any attempt to solve this led me to drown in several equations … Is there a trick I can not see? That's wrong? This comes from an elaborate problem that would take a long time to explain, so maybe it is not true and I made a mistake in my ~ 15 page manuscript. An answer to this question would allow me to conclude (if it is true) some tedious test of a result unrelated to complex algebra.

Any ideas or suggestions?