4 problems

Let $P$ be a lattice polytope, let $k[P]$ be its polytopal algebra, and let $f:k[P]\to k[P]$ be a codimension $1$ retraction. Write $f^{\alpha}=\alpha\circ f\circ\alpha^{-1}$ for c…

Let $k[P]$ be a polytopal algebra, and let $f:k[P]\to k[P]$ be a retraction, meaning an idempotent homogeneous endomorphism. Image conjecture. The image of every retraction in…

Let $k$ be a field and let $\operatorname{Pol}(k)$ denote the category of polytopal algebras over $k$. A homomorphism is called tame if it is obtained from the identity mapping…

Tameness conjecture. Any homomorphism in $\operatorname{Pol}(k)$ is obtained by a sequence of these constructions, starting from the identity mapping $k\to k$. Moreover, there are…