Equation of the type polynomial${}= \bar{z}$ Let $P(z)$ be a complex polynomial of degree 3. How many roots equation $P(z) = \bar{z}$ could have?
I had tried the following ideas but was unable to push through:


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*Research the ring $\mathbb{C}[X]\otimes \mathbb{C}[\bar{X}]$ and ideals of it that annul specific amount of points. But polynomials like $X - \bar X$ had spoiled this idea, maybe it would be better if we will limit ourselves to polynomial with different inversion degree and straight degree but no idea how to approach it further

*Try to find out how polynomials twist the space and find amount of points that are gluing together and their image is a symmetry relative to real line, i had stopped at trivial case $aX^3$ and it was hard to add another degree.

*All polynomials of degree 3 are path-connected in $\mathbb{C}$. Maybe I can find out that amount of solutions of $P(z) = \bar z$ is some nice characteristic of polynomials that have some properties to use. But this one was quite desperate measure already.

 A: Sketch of the answer with references as the problem is non-trivial even in degree $3$; by the general theory of harmonic polynomials of degree $n$, so
$P(re^{it})=\sum_{-n \le k \le n}a_kr^{|k|}e^{ikt}, a_n \ne 0 \lor a_{-n} \ne 0$ we have a fundamental theorem in the non-degenerate case $|a_{-n}| \ne |a_{n}|$, namely
$P$ has only isolated zeroes, one can define an integral multiplicity at each and the number of zeroes counted with the absolute value of their multiplicity is between $n$ and $n^2$ and there are examples that show both end possibilities are attained. 
However, in the case where $P=\bar z+T$ with $T$ analytic polynomial of degree $n \ge 2$ Wilmhurst conjectured and Khavinson and Swiatek proved (link to pdf paper in Proc Amer Math Soc) that the number of solutions is at most $3n-2$ (which is attained for $n=3$ with a nice example there with $7$ roots)
So summing, the OP case  answer is known to be between $3$ and $7$ with all attained
For the general theory and proofs of the general results above (Bezout theorem and a version of the argument principle for continuous functions in the plane), the book Complex Polynomials by T. Sheil-Small (which I highly recommend for anyone interested in the topic) has the details in chapters 1 and 2
The survey of Khavinson et al from 2018 has more on the topic of zeroes of harmonic polynomials.
