24 problems

Schinzel–Zassenhaus conjecture. There exists a constant $c>0$ such that, if $\alpha$ is not a root of unity, then

Let $\alpha$ be a non-zero algebraic integer of degree $n$ that is not a root of unity, and define its house by … where the $\alpha_k$ are the conjugates of $\alpha$. An algebraic…

Let $s$ be the growth number of the real cubic family, and consider a constant-growth-number locus in the parameter plane. Algebraic-integer conjecture. If this locus has interior…

Let $\xi\in\mathbb{R}$ be neither rational nor quadratic over $\mathbb{Q}$, and let $H(\alpha)$ denote the height of an algebraic number $\alpha$. Conjecture on algebraic-integer a…

Let $\alpha$ be a cyclotomic integer, meaning an algebraic integer expressible as a sum of roots of unity. Let $l(\alpha)$ be the smallest number of roots of unity occurring in any…

For each integer $n\geq 1$, let $\mathcal{A}_n$ be the set defined in the paper, viewed as a subset of the algebraic integers $\overline{\mathbb{Z}}$. A subset is polynomially clos…

Let $\alpha$ be a nonzero algebraic integer that is not a root of unity, with all of its conjugates lying in the sector … where $0<\theta<90^\circ$. Let $c(\theta)>1$ be the functi…

Let $\mathcal{C}$ be a spherical fusion category over $\mathbb{C}$, and let $s\geq 0$ be rational. If $A$ is its fusion ring, a category is $s$-Isaacs when, for every character…

Let $\mathcal{F}$ be a complex commutative Grothendieck ring with simultaneously diagonalized fusion matrices having eigenvalues $(\lambda_{i,j})$. Let $(c_j)$ be the eigenvalues o…

Integrality conjecture. $cech{\mathsf{C}}{alpha}$ is always an algebraic integer.

Dinh–Fornæss conjecture. The number $\lambda(f)$ is an algebraic integer of degree at most $n$; if $f\in\operatorname{Aut}(\mathbb{A}^n_{\mathbf{k}})$, its degree is at most $n-1$.

A totally positive algebraic integer is an algebraic integer all of whose algebraic conjugates are positive real numbers. Totally positive algebraic integer Laplacian-eigenvalue co…

Let $k\in\mathbb{N}$ and let $\lambda_1,\ldots,\lambda_k\in\mathtt{TRAI}\setminus\{0\}$. For each $\lambda_i$, let $\mathcal{T}_{\mathrm{min},\lambda_i}$ denote the set of trees ha…

Let $d$ be an even natural number, and let $P_d(x)$ be the minimal polynomial of an extremal reciprocal algebraic integer of degree $d$. A polynomial is primitive in the sense used…

Let $\alpha$ be a reciprocal algebraic integer that is not a root of unity, and let $d=\deg(\alpha)\geq 2$. Let $\mathrm{mr}^d(d)$ denote the relevant degree-$d$ extremal reciproca…

Schinzel–Zassenhaus–Boyd conjecture. There is a constant $T>1$ such that if $\alpha$ is not a root of unity, then

Composite-degree extremal reciprocal polynomial conjecture. If $R_{2p_i}(x)$ is the minimal polynomial of the extremal reciprocal algebraic integer of degree $2p_i$, then

Let $d\geq 17$ be prime with $d\equiv 5\pmod 6$. Define … Let an algebraic integer of degree $d$ that is not a root of unity be extremal if it has the minimum possible house among…

Let $\mathrm{mr}(d)$ be the minimum of the houses of reciprocal algebraic integers of degree $d$ that are not roots of unity. An algebraic integer attaining $\mathrm{mr}(d)$ is cal…

Hoffman's conjecture. Every totally real algebraic integer is an eigenvalue of a graph.

A real algebraic integer $\lambda$ is representable if there exists an integral symmetric matrix $M$ such that the map $\lambda\mapsto M$ is an isomorphism of algebras…

Work over $\mathbb{C}$. For a totally real algebraic integer $\lambda$ of degree $d>1$, let $E_s(n,\lambda)$ be the maximum multiplicity of $\lambda$ as an eigenvalue of an $n$-ver…

Bruinier and Ono's integrality conjecture. For the Maass form $P_p(z)$ and the CM points $\alpha_Q$ occurring in the formula for $p(n)$, the number

An algebraic integer is a complex number that is a root of a monic polynomial with integer coefficients. A chromatic root is a zero of the chromatic polynomial of a finite graph. S…