22 problems

Let $G$ be a finite $p$-group, let $k$ be a positive integer, and let $d$ and $a_1,\dots,a_d$ be integers. A regular sequence has degree sequence $a_1,\dots,a_d$ when its elements…

Let $G$ be a finite $p$-group. Its cohomology ring $H^*(G,\mathbb F_p)$ is a graded-commutative, noetherian, local $\mathbb F_p$-algebra. Write $\operatorname{rank}_p Z(G)$ for the…

Eisenbud–Green–Harris conjecture. The ideal $I$ has the same Hilbert function as an ideal containing $x_1^{a_1},\dots,x_n^{a_n}$. This conjecture predicts that the Hilbert function…

Let $K$ be a field of characteristic zero, let $S=K[x_1,x_2,x_3,x_4]$, and write $p_m=x_1^m+x_2^m+x_3^m+x_4^m$. Let $A=\{a_1,a_2,a_3,a_4\}$ be a set of positive integers with…

Let $K$ be a field of characteristic zero, let $p_m=x_1^m+x_2^m+x_3^m$, and let $a,b,c$ be integers satisfying $0<a<b<c$ and $\operatorname{gcd}(a,b,c)=1$. The 6abc conjecture. The…

The v-function formula. For all $k\ge1$,

Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $M$ be a finitely generated unmixed $R$-module. Let $\mathbf{x}=x_1,\ldots,x_m$ be an amenable partial system of parameter…

Let $k,\ell\in\mathbb{N}$ be multiplicatively independent, meaning that no positive powers of $k$ and $\ell$ are equal. Let $\mathbf a=(a_n)_{n\in\mathbb{N}_0}$ be an integer seque…

Let $V$ be a vector subspace of $\bigoplus_{i=1}^\infty \mathbb{Q}$. For a prime $p$, let $\nu_p$ denote the $p$-adic valuation, and let $p_1,p_2,\dots$ be the increasing enumerati…

Regular-sequence conjecture. The displayed collection is a regular sequence on $R/I$ and hence is part of a system of parameters on $R/J$ and $R/Q$.

Let $\mathfrak{g}$ be a reductive Lie algebra, let $e$ be a nilpotent element, and write $\mathfrak{g}^{e}$ for its centralizer. Let $\mathcal{F}_\xi(\mathfrak{g}^{e})$ be the Mish…

Let $k\geq 2$, let $\mathpzc{k}$ be the coefficient field, and let $f(z)\in\mathpzc{k}[[z]]$ be a Mahler power series. A sequence of rational functions is calm in the sense defined…

Let $k\geq 2$, let $\mathpzc{k}$ be the coefficient field, and let $f(z)\in\mathpzc{k}[[z]]$ satisfy the Mahler equation … A sequence of coefficients is precalm in the sense define…

Let $k\geq 2$, let $\mathpzc{k}$ be the coefficient field, and let $f\in\mathpzc{k}[[z]]$. A rational function $h(z)\in\mathpzc{k}(z)$ is $k$-regular if it is a $k$-regular rationa…

Conca–Krattenthaler–Watanabe's conjecture. The polynomials $p_a(3),p_b(3),p_c(3)$ form a regular sequence in $S$. Conca et al. verified some special cases, but the conjecture remai…

Let $H$ be a subgroup of $GL_n({\mathbb F}_q)$ acting on $S={\mathbb F}_q[\mathbf{x}]$, and let $D_{n,j}$ denote the Dickson polynomials. Landweber–Stong conjecture. The depth of t…

$j_1$ regular-sequence conjecture. If

Let $A\subset\mathbb{N}^*$ have $|A|=4$, say $A=\{a_1,a_2,a_3,a_4\}$, with $\gcd(A)=1$, and let $p_A(4)$ denote the corresponding four power sum symmetric polynomials in four varia…

Let $S=\mathbb{C}[x_1,x_2,x_3,x_4]$. The prime-ideal conjecture for pairs of power sums. The following assertions hold: (1) if $a$ is prime, $a\ge5$, and $b=a+m+6d$ with…

Let $p_i$ be the power sum symmetric polynomial of degree $i$ in $S=K[x_1,x_2,x_3,x_4]$. Let $A=\{a,b,n\}$ with $a<b<n$. The four-variable power-sum conjecture. The sequence…

Let $F_n$ denote the $n$th Fibonacci number, let $up$ be the $p$-adic valuation, and let $\alpha(p)$ be the least positive integer $n$ such that $p$ divides $Fn$. For a sequence, i…

Let $h_k(3)$ be the complete symmetric polynomial of degree $k$ in three variables, and for $A=\{a,b,c\}$ with $a<b<c$ let $h_A(3)$ denote the set of polynomials…