31 problems

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…

Regular-sequence conjecture. The elements $e^0,e^1_{m,n,k},\ldots,e^{mn}_{m,n,k}$ form a regular sequence in $K[(\Delta_m)^{\times n}]$ of maximal length.

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…

Becker's conjecture. If $F(z)$ is a $k$-regular power series, then there exists a nonzero $k$-regular rational function $R(z)$ such that $F(z)/R(z)$ satisfies a Mahler-type functio…

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…

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 $k\ge1$,

Regular-sequence formulation of the Casas–Alvero conjecture. For every $i\in\{1,\ldots,d-2\}$, $R_{i+1}$ is not a zero divisor modulo $(R_1,\ldots,R_i)$; equivalently,…

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…

Let $\mathcal{A}=\{a,b,c\}\subset\mathbb{N}^*$, and call $\mathcal{A}$ associated to a regular sequence if the system of diagonal equations … has only the trivial solution…

A rational base numeration system has base $\frac{p}{q}$, and it is expanding when the base satisfies the expanding property discussed above. The preceding results concern, in part…

Let $S_d$ be the space of binary forms of degree $d$, let ${\rm Inv}_d$ be its ring of $SL_2$-invariants, and, for $d=2k$ with $k$ even and $n\ge k$, let…

Base-3 recurrence conjecture. For all $n\geq 0$,

Conjectured recurrences. For all relevant nonnegative integers $n$, the discriminator sequence satisfies

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 $f(z)\in\mathpzc{k}[[z]]$. A power series is $k$-Dumas if it satisfies a Mahler equation whose rational-function…

Let $k\ge 2$ and let $\ell\ge 1$ be integers. A sequence is $k$-automatic if it is generated by a finite automaton reading the base-$k$ representation of its indices. For an infini…

Let $k\geq 2$ be an integer, and let $w$ be a $k$-automatic infinite word. Its $2$-abelian complexity is the sequence counting the equivalence classes of factors of each length und…

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