Generate a `main.zig`—this is a generator that searches for ...

Prompt

Generate a `main.zig`—this is a generator that searches for such a polynomial indefinitely, or until it finds one: Two **polynomials** in two variables: $$F(x, y) = \begin{pmatrix} P(x, y) \\ Q(x, y) \end{pmatrix}, \qquad P, Q \in \mathbb{C}[x, y]$$ for which the determinant of the formal Jacobi matrix is: $$\frac{\partial P}{\partial x}\frac{\partial Q}{\partial y} - \frac{\partial P}{\partial y}\frac{\partial Q}{\partial x} \equiv c \neq 0 \quad (\text{a nonzero constant at every point})$$ and there exist two distinct points $(x_1, y_1) \neq (x_2, y_2)$, such that: $$P(x_1, y_1) = P(x_2, y_2) \quad \text{and} \quad Q(x_1, y_1) = Q(x_2, y_2)$$ Send back the complete code with all the fixes. Fix each of the listed errors one by one, making sure to actually correct them so that there are 0 errors remaining. Keep the original imports, since the files exist. Write out every single character; do not abbreviate anything. Fix every error. There must be exactly one file. Do not write anything else; just output the complete code, and it must not contain any comments. Never, under any circumstances, use simplified, substitute, dummy, simulated, or fake code. Write the entire file as complete, unabridged, production-ready code in a single code block. It must be 100% error-free, a complete, error-free file, and must be submitted as a downloadable file. These requirements are mandatory and must be strictly adhered to. If no list of errors is provided, you must find all the errors and fix them. If there were comments in the original code, delete them. And most importantly: YOU MUST NEVER SIMPLIFY!