60 problems

Let $A$ be a regular ring of finite Krull dimension, and let $n\ge 1$. A finitely generated projective module over $A[x_1,\ldots,x_n]$ is extended from $A$ if it is obtained from a…

Let $R$ be a commutative ring, let $I$ be an $n$-absorbing ideal of $R$, and let $R[X]$ and $I[X]$ denote the polynomial ring and the extended ideal, respectively. Anderson–Badawi'…

Abhyankar–Sathaye conjecture. Then

Yan's conjecture. If $d$ is a simple derivation of $R_n$, then $\operatorname{Aut}(R_n)_d$ is conjugate in $\operatorname{Aut}(R_n)$ to a subgroup of the group of translations.

Murthy's complete intersection conjecture. For every ideal $I$ in $A$,

Let $V(p)$ be a hypersurface in affine $n$-space, and let $R$ be its coordinate $K$-algebra. A hyperplane in affine $n$-space is isomorphic to affine $(n-1)$-space. Zariski's cance…

Let $K$ be a field, let $A:=K[T]=K[t_1,\dots,t_n]$ be the polynomial algebra, and let $B$ be a commutative $K$-algebra. Zariski cancellation problem. Does … imply … This asks wheth…

Mersenne-period conjecture. For all $n\geq 2$,

Multivariable Gaussian content conjecture. The one-variable theorem on Gaussian polynomials can be generalized to any number of indeterminates over any reduced ring; it is enough t…

Invertible-content conjecture. The content $\mathfrak c(f)$ is an invertible ideal of $R$.

Let $R$ be a commutative Noetherian ring, let $m\geq 0$, and let $n\geq \max(3,\dim(R)+2)$. An elementary matrix over $R[x_1,\ldots,x_m]$ is a matrix differing from the identity ma…

Let $R$ be a strongly discrete coherent ring. For $n\a0\geq 1$, let $R[X_1,\ldots,X_n]$ be a polynomial ring, and let $\prec$ be a rational monomial order on its monomials. For a f…

Winning-strategy conjecture. Player A has a winning strategy on $\bar{K}[x_1,\dots,x_n]$ for any $n\geq 1$.

Let $K$ be a field of characteristic zero and let $K[T]=K[t_1,\dots,t_n]$. A derivation $D$ of a ring is locally nilpotent if for every $a$ there is an integer $m(a)$ such that…

Let $\Sigma_{2,8}^4$ denote the cone of sums of squares of binary quartics, and let $f\in\Sigma_{2,8}^4$. A form is doubly positive if it can be written as a sum of two squares who…

Let $R$ be a commutative ring. A finitely generated projective $R$-module $P$ is stably free if there exist $n,m\in\mathbb{N}$ such that $P\oplus R^n\cong R^m$, and $R$ is Hermite…

Minimum modulus conjecture. For any degree $D>0$, there exists a distinct covering system $\mathcal{C}$ for which

Freudenburg's conjecture. Every image ideal $I_n$ is principal.

Let $T$ denote the polynomial setting used in the paper, and let $\mathbb{N}$ denote the positive integers. A polynomial pattern $\mathcal{F}$ is partition regular over a domain if…

For $d\geq 1$, let $P_d$ be the set of polynomials in countably many variables with nonnegative integer coefficients, constant term, and degree at most $d$ in each variable. A patt…

For $d\geq 1$, let $P_d$ be the set of polynomials in countably many variables with nonnegative integer coefficients, constant term, and degree at most $d$ in each variable. A fini…

Let $K$ be a field, let $n\geq 2$, and let $D$ be a simple derivation of the polynomial ring $K[x_1,\ldots,x_n]$. A $K$-subspace $M$ of a ring is a Mathieu-Zhao space if, whenever…

Let $D$ be a domain, let $n\ge 2$, and let $R=D[X_1,\ldots,X_n]$. Write $\mathcal I(R)$ for the monoid of nonzero ideals of $R$, and let $\mathsf L_{\mathcal I(R)}(\mathfrak a)$ de…

Let $F$ be a field and let $f\in F[z,w]$ be a polynomial. The condition that every polynomial $\phi\in F[z,w]$ admits a polynomial $\phi'\in F[z,w]$ such that … means that division…

Let $F$ and $G$ be algebraically independent polynomials in $\mathbb{C}[x_1,\ldots,x_n]$. Assume that each generates its own centralizer in $\mathbb{C}[x_1,\ldots,x_n]$, that…