# 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?