20 problems

Let $W_{d^2}[n]$ be the locus under consideration, and for odd $n$ let $W_{d^2}^{\epsilon}[n]$, with $\epsilon\in\{0,1\}$, denote its two parity-labelled subloci. Parity conjecture…

Let $P_n(z)$ be the sorted binomial polynomial with parameter $n$. Irreducibility conjecture. $P_n(z)$ is irreducible over $\mathbb{Q}$ for all even $n$. The analogous factorizatio…

Let $k$ be a field of characteristic $p\geq 3$, and let $\mathcal{H}_g$ be the moduli space of smooth hyperelliptic curves of genus $g$ over $k$. Denote by $\mathcal{H}_g^{g-2}$ th…

Let $k$ be a field of characteristic $p\geq 3$, and let $\mathcal{H}_g$ be the moduli space of smooth hyperelliptic curves of genus $g$ over $k$. Write $\mathcal{H}_g^{g-1}$ for it…

Let $K$ be a field and let $b\in K((\mathbb{R}^{\leq 0}))$ be a random generalized power series, meaning that either $\operatorname{cl}(\operatorname{supp}(b))-\{0\}$ is…

Let $f_{a,b}(z)=z^{3}-3a^{2}z+2a^{3}+b$ be a monic, reduced cubic polynomial with marked critical point $a$, and let $\a7\Sigma_{k,n}$ be the Zariski closure in the moduli space…

Irreducibility conjecture. If $p\nmid a_{n}a_{q}$, $q\nmid u$, and

Stability conjecture. If $f(z)=z^d+\frac{1}{c}$ is irreducible over $\mathbb{Q}$, then $f(z)$ is stable over $\mathbb{Q}$.

Merino's irreducibility conjecture. This polynomial is irreducible over $\mathbb{R}$ if and only if $m$ and $n$ are coprime. The claim gives a precise arithmetic criterion for irre…

Irreducibility conjecture. If $|a|\geq 5$, then $x^{n}+ax^{m}\pm1$ is irreducible over $\mathbb{Q}$. Furthermore, there are only finitely many reducible polynomials of the form…

Irreducibility conjecture. The polynomial $R_n(x;\ell_1,\ell_2,y)$ is irreducible unless $\ell_1=\ell_2$ and $n$ is odd. In that exceptional case, $R_n(x;\ell_1,\ell_2,y)$ is the p…

Let $S evarnothing$ be a zero pattern, let $Uinmathbb{R}^{mtimes n}$, and let $rin[m-1]$. Denote by $Z{U,r}^{S}$ the critical-point variety and by $HU^{S}$ the corresponding affine…

Let $Q_n^{(k)}(x)$ and $V_n(x)$ be the polynomials defined in the paper, with integers $n\geq 1$ and $0\leq k\leq n-1$. Irreducibility conjecture. For all integers $n\geq 1$ and…

Toric pullback irreducibility conjecture. There is a finite union $\mathcal{E}$ of proper subtori of $\mathbb{G}_{\rm m}^n$ and a finite set $\Lambda$ of isogenies of…

Let $K$ be a field of characteristic different from $2$, and let $f(x)\in K[x]$ be a monic, post-critically finite quadratic polynomial. Let $o_f$ denote the period of the post-cri…

Let $S$ be the set given in Proposition 3.8. For $m\in S$, define … Stability conjecture. For every $m\in S$, the polynomial $f$ is stable, meaning that every iterate $f^n$ is irre…

Let $K$ be a field, let $\phi\in K(z)$ have degree $d\geq 2$, and let $\alpha\in\mathbb{P}^1(K)$ be not periodic under $\phi$. For a point $\alpha\ne\infty$, write $(\phi,\alpha)$ a…

Call an eta quotient primitive if it is not a rescaling of another eta quotient by a positive integer. For an eta quotient $f$, its extract is the primitive eta quotient $f_0$ of w…

Let $f$ be an irreducible holomorphic eta quotient, and for a positive integer $\nu$ let its rescaling be $f_\nu(z)=f(\nu z)$. Rescaling irreducibility conjecture. The rescalings o…

Let $f$ be a holomorphic eta quotient of level $N$. It is quasi-irreducible if it is not factorizable on $\Gamma_0(N)$, whereas it is irreducible if its only factors are $1$ and…