4 problems

Let $V\subseteq\mathbb{C}^n\times(\mathbb{C}^{\times})^n$ be an algebraic variety, and let $\pi_1$ and $\pi_2$ be the projections to the additive and multiplicative factors. Call…

Gallinaro's conjecture. Such a decomposition should be possible with the $C_i$ taken to be semi-algebraic.

Abelian Exponential-Algebraic Closedness conjecture. If $V$ is a free and rotund algebraic subvariety of $\mathbb{C}^g\times A$, then

Let $k\subset\mathbb{C}$ be a finitely generated field, and let $p(x,y)\in k[x,y]$ be irreducible with $\frac{\partial p}{\partial x},\frac{\partial p}{\partial y}\neq 0$. Zilber's…