97 problems

For integers $n>2$ and $k>1$, let $F_{n,k}(x)=\Phi_n(1+x+\cdots+x^k)\in\mathbb{Q}[x]$, where $\Phi_n$ is the $n$-th cyclotomic polynomial. Determine, exactly in terms of $n$ and…

Let $\varphi=\frac{1+\sqrt{5}}{2}$ be the golden ratio, and let $\langle\varphi\rangle=\{k\varphi:k\in\mathbb{Z}\}$ be the subgroup of $\mathbb{C}$ generated by $\varphi$. An irred…

Rational Stark conjecture. There exists an element $u_\chi\in(k\otimes_{\mathbb{Z}}U_{v,S})^{\chi^{-1}}$ such that

Fix an integer $d \geq 3$. For a totally real number field $K$ of degree $d$, let $\operatorname{disc}(K)$ denote its discriminant, and let an ideal class mean a class of fractiona…

Oppenheim's conjecture. The lattice

Let $N$ be a prime level, let $\chi$ be the character used to define $h_\chi$, and let $B_{m,\chi}$ denote the $m$-th generalised Bernoulli number attached to $\chi$. Write … For e…

Algebraicity–multiplier group conjecture. A quasiperiodic flow $\psi$ is $F$-algebraic if and only if $M_\psi$ is a finite index subgroup of ${\mathfrak o}_F^*$.

Let $n\geqslant 3$ and let $\Lambda\subset\mathbb{R}^n$ be an $n$-dimensional lattice with positive norm minimum $N(\Lambda)>0$. A lattice is called algebraic when it is similar, m…

Let $p$ be an odd prime splitting completely in $k$, let $K/k$ and the other notation be as in the paper, and let $G=\operatorname{Gal}(K/k)$. Let $\eta^+_{K/k}$ be the image of th…

Let $K^+/k$ be the totally real subextension under consideration, let $G^+=\operatorname{Gal}(K^+/k)$, let $S_0=S_{\rm ram}(K/k)$, and let $U_{S_0}(K^+)$ be the group of global…

Let $k$ be a totally real number field, let $p$ be an odd prime splitting completely in $k$, and let $K$ be as in the paper. Write $G=\operatorname{Gal}(K/k)$ and let $\delta(K/k)$…

Let $\phi$ be a quasiperiodic flow on $T^n$ generated by a constant vector field $X$. Call $\phi$ algebraic if it is $F$-algebraic for a real algebraic number field $F$ of degree…

Let $F$ be a real algebraic number field of degree $n$ over $\mathbb{Q}$, and let $T^n$ denote the $n$-torus. The multiplier-group equality conjecture. There exists a quasiperiodic…

Let $T^n$ be the $n$-torus, let $\phi$ be a quasiperiodic flow on $T^n$ generated by a constant vector field $X$, and let $F$ be a real algebraic number field of degree $n$ over…

Let a SIC-POVM in $\mathbb{C}^d$ have normalized overlaps $e^{i\theta_{jl}}$, where the angles $\theta_{jl}$ are defined by the normalized inner products of its fiducial vectors. A…

Gras's conjecture. For $p\gg 0$, one has

Let $p$ be the odd prime and let $\theta$ be the algebraic number considered in the preceding discriminant computation, with $\mathbb{Q}(\theta)$ its generated number field. Discri…

Outer-derivation conjecture. Under the hypotheses of the determinant conjecture, $D$ is outer if and only if

Let $\mathbb{K}=\mathbb{Q}(\zeta)$ be the $p^k$th cyclotomic number field, where $k\in\mathbb{N}\setminus\{1\}$ and $p$ is a rational prime, and let…

Innerness conjecture. Under the hypotheses of the determinant conjecture, if $1\le e_1\le r-1$ and $e_2\ge1$, then $D$ is inner if and only if

Let $\mathbb{K}=\mathbb{Q}(\zeta)$ be the $n$th cyclotomic number field with $n=2^r p$, where $r\in\mathbb{N}$ and $p$ is an odd rational prime, and let…

Let $\beta\in\mathbb R$ satisfy $a\beta^2+b\beta+c=0$ with $a,b,c\in\mathbb Z$ and $b^2-4ac$ nonsquare, and let $\r\in\mathbb Q^2$. Let \shin^{\r}[\beta] and…

Let $a,b\in\mathbb{Z}$ with $a\neq 0$ and $b\geq 1$, and let $m_{a,b}:=a^2+ab+b^2$. Using the polynomials and Laurent polynomials defined in Proposition 19, let $G_{a,b,m}(T)$ and…

For $d\geq 1$, define … Let $a,b\in\mathbb{Z}$ be nonzero, put $c:=a+b\neq 0$, and define … For $d\in\{a,b,c\}$, … If $a$ is even and $b$ is odd, also define … Then … This identity…

Fix a positive integer $m$. For an algebraic integer $\alpha\in\overline{\mathbb{Q}}$, define … where $M_{\sigma(K)}(\alpha)$ is the relevant relative Mahler measure; for $m=1$, th…