25 problems

Nowicki's conjecture. The algebra $K[X_n,Y_n]^\delta$ is generated by $x_1,\ldots,x_n$ and

Let $k$ be a commutative ring, let $k[t]$ be the polynomial algebra, and for $m\geq 0$ let $HS_1(k[t])_{t^m}$ denote its $t^m$-layered symmetric homology, obtained from the decompo…

Retract conjecture. If the corresponding Jacobian matrix of $p$ and $q$ is invertible, then $K[p]$ is a retract of $K[x,y]$.

One-singular-reduction conjecture. If $p\in P_2$ has a unimodular gradient, then one can get

Let $d\geqslant 1$ be the dimension of the base $\mathbb{R}^d$, let $k\geqslant 1$ be a differential order, and set … Let $W_d^k$ denote the Wronskian $N$-ary bracket on…

A quadratically superintegrable system is a superintegrable system whose integrals of motion are differential operators of order at most two; its algebra of integrals is the algebr…

Let $k$ be an integer. For the TTW quantum system, let $H$ be the Hamiltonian, let ${\cal I}_{1}$ and ${\cal I}_{2}$ be its two integrals, and set … The polynomial algebra of integ…

Let $k$ be an integer, and consider the TTW quantum system on the plane with Hamiltonian $H$ and two integrals ${\cal I}_{1}$ and ${\cal I}_{2}$, together with their commutator…

Let $N$ be a positive integer and let $\gamma_k$ be coefficients satisfying $1\leq k\leq N$. Let $\hat{\mathcal{H}}_N$ be the quantum Hamiltonian, let $\hat{\mathcal{C}}$ be the qu…

Let $P_k=\mathbb F_2[x_1,\ldots,x_k]$ be the polynomial algebra, let $Q P_k$ denote its quotient by the image of the positive-degree Steenrod algebra, and let … be Kameko's squarin…

The order conjecture. For arbitrary $N$, the polynomial algebra above is of $(2N-1)$th order; for $N=2$, it reduces to the well-known cubic Higgs algebra of the proper Zernike syst…

Higher-order derivation conjecture. If

Intersection-derivation conjecture. The set of derivations of $A$ is the union of the derivations of $A_1$ and $A_2$, restricted to $A$. This conjecture concerns how derivations be…

Let $A\subseteq\mathbb{K}[x]$ be a finite-codimension subalgebra, let $k_\alpha$ be the dimension of the vector space of all $\alpha$-derivations of $A$, and let…

Let $A\subseteq\mathbb{K}[x]$ be a subalgebra with spectrum containing $\alpha$, let $C=\{\alpha_1,\ldots,\alpha_s\}$ be its spectrum, and let $\alpha_j\sim\alpha$ mean that…

Let $A=\mathbb{K}[p,q]$ be the subalgebra generated by $p$ and $q$, let $\chi_{p,q}(x)$ be their characteristic polynomial, and let $a(x)$ be a polynomial of degree at least two di…

Let $R\colon \mathbb{R}[x]\to\mathbb{R}[x]$ be an injective Rota–Baxter operator of weight zero. A formal integration at some point is the corresponding formal integration operator…

Freudenburg's conjugacy conjecture. Every maximal Lie subalgebra $\mathcal{A}\subset\operatorname{LND}(\mathbb{K}[x_1,\ldots,x_n])$ is conjugate to $\mathfrak{T}$. This is the uniq…

Let $\Bbbk[x]$ be the polynomial algebra in one variable, and let … be the collection of injective Rota–Baxter operators of weight zero described in the source. Classification conj…

Higher-rank polynomial algebra conjecture. Symmetry algebras of $n$-dimensional superintegrable systems would in general be higher-rank polynomial algebras, although only a few iso…

Let $A_n=\mathbb{C}[x_1,\ldots,x_n]$, and let $f_1,\ldots,f_n\in A_n$. Define the Jacobi determinant … A tuple with $J(f_1,\ldots,f_n)\in\mathbb{C}^\times$ is called a Jacobi tuple…

Let $m>3$, let $\mathfrak{R}_{m,\omega}$ be the random graph with colours $c_0,\ldots,c_{m-1}$, with $c_0$ transparent, and let $H=\operatorname{Sym}(m-1)$ act on the remaining col…

Let $B\subset\mathbb R^n$, let $w(z)$ be a weight function, and let $\{u_\alpha\mid\alpha\in\mathbb N^n\}$ be a sequence of orthogonal polynomials over $B$. Let $\mathcal M$ be the…

Let $z=(z_1,\ldots,z_n)$ be commuting variables and let $\mathbb C[z]$ be their polynomial algebra. For $1\leq i\leq n$, write $\partial_i=\partial/\partial z_i$. Let…

Automorphic-orbit conjecture. Let $p\in A_n-K$. Then any endomorphism of $A_n$ preserving the automorphic orbit of $p$ must be an automorphism.