12 problems

Let $k$ be a field of characteristic zero and let $\operatorname{Aut}_k k[x_1,x_2,x_3]$ be the automorphism group of the polynomial ring in three variables. Define…

Abhyankar–Sathaye conjecture. Then

Let $k$ be a field, let $x$ be an indeterminate, and let $f\in k[x]^{[n]}$ be a residual coordinate, meaning that it becomes a coordinate modulo $x-c$ for each $c\in k$. Dolgachev–…

Abhyankar–Sathaye embedding conjecture. If $H$ is isomorphic to a coordinate hypersurface, then $H$ is equivalent to it under an automorphism of the ambient affine space.

Residual-variable conjecture. If $p$ is a residual $x_1$-variable, then $p$ is an $x_1$-variable, meaning

Shastri's conjecture. The algebraic trefoil knot embedding is non-rectifiable over $\mathbb{C}$.

Interpolation conjecture. Every smooth connected affine algebraic curve in ${\mathbb C}^2$ is interpolating for any Bargmann–Fock space.

Extension conjecture. Data from $C_2$ could perhaps be extended to $S$.

Let $\mathcal I_{\mathbb R^4}$ be the moduli space of instantons on $\mathbb R^4$, and let $\mathcal I_{\mathrm{TN}}$ be the moduli space of instantons on a Taub–NUT space. Instant…

Let $k\geq 2$. Consider the forcing algebras associated with the $k$th powers of the equation $xv+yu=z(1-z)$; these are affine torsors over $D(x,y,z)$ whose products with…

Let $K[x_1,x_2,x_3]$ be a polynomial algebra in three variables. An affine automorphism is an automorphism induced by an invertible affine-linear change of the variables, and an ex…

A polynomial embedding is an injective polynomial map whose image is isomorphic to the complex line. Two polynomial embeddings $\phi_1,\phi_2:\mathbb{C}\hookrightarrow\mathbb{C}^3$…