49 problems

Let $\chi$ be a multiplicative character of order $c$ over a finite field, let $f(x)=x^d+\lambda x^e$ with $(d,e)=1$, and let $p$ be the characteristic. Write…

Let $\mathcal{R}=\mathbb{C}[x,y]$, and let $N^0(F)$ denote the Newton polygon of $F$ with the origin as reference point. Let $a,b\in\mathbb{Z}_{>0}$ be relatively prime with…

Let $\Delta$ be a lattice polytope of degree denominator $D$, and let ${\rm GNP}(\Delta,p)$ and ${\rm HP}(\Delta)$ denote its generic Newton polygon and Hodge polygon, respectively…

Let $f$ be a non-constant monic polynomial in $Q[x]$. A global permutation polynomial (GPP) over $Q$ is a polynomial $P(x)\inQ[x]$ whose reduction modulo $p$ is a permutation of…

Let $C_f(s,T)$ be the $C$-function of the $T$-adic exponential sums associated with a polynomial $f$, and let its absolute $T$-adic Newton polygon mean the $T^a$-adic Newton polygo…

Let $\Delta$ be a triangle with vertices at $(0,0)$, $\mathbf P_1=(a_1,b_1)$, and $\mathbf P_2=(a_2,b_2)$. Assume the hypotheses on the prime $p$ from the main theorem. Let…

Wan's conjecture. There is a Zariski dense subset ${\mathcal{U}}$ in $\mathbb{A}^d$ such that, for every $f(x)\in {\mathcal{U}}$, the limit exists and

Saturation conjecture. The terms that appear in the numerator of a Markov polynomial $M_{\rho}$ are precisely those corresponding to the integer lattice points in the Newton polygo…

Let $p$ be a prime, and let $\xi_1$ and $\xi_2$ be Newton polygons that are realized in characteristic $p$. Oort's direct-sum conjecture. Then $\xi_1 \oplus \xi_2$ is realized in c…

Let $g \geq 18$, and let $m$ be the least common multiple of the exponents of all of the Galois groups of abelian covers of $\mathbf{P}^1$ branched at three points of genus $g$. Un…

Let $\rho=a/b$ be rational, and let the critical triangle be the set of lattice points satisfying … Consider the coefficients of the Markov polynomial associated with these lattice…

Saturation conjecture. The terms appearing in the numerator of $M_\rho$ are precisely those corresponding to the integer lattice points in $\Delta_\rho$. The conjecture is an equiv…

Let $\overline{\mathcal{R}}=\mathbb{C}[x^{1/\mathbf d},y]$ for a fixed positive integer $\mathbf d$, and let $\overline{T}_{m,n,a}$ be the class defined in the paper. Suppose…

Let $Z$ be the innermost polynomial and $\zeta(F)$ the inner polynomial arising from the paper's construction. For a convex polygon, the NE vertex is the uppermost vertex with larg…

Let $\mathcal{Q}$ and $R_{m,n}$ be as above, and let $\zeta(F)$ denote the inner polynomial of $F$. The inner-polynomial lattice-point conjecture. If $(a,b,m,n)\in\mathcal{Q}$ and…

The structured Newton-polygon conjecture. Suppose $(a,b,m,n)\in\mathcal{Q}$ and $F,G\in\mathcal{R}$ satisfy $[F,G]\in\mathbb{C}$, $F\in R_{m,n}$, and $G\in R_{bm/a,bn/a}$. Then…

Generalized Dumas–Eisenstein irreducibility conjecture. The polynomial

Oort's conjecture. If $\xi_i$ occurs on $\mathcal{M}_{g_i}$ for $i=1,2$, then $\xi_1\oplus\xi_2$ occurs on $\mathcal{M}_{g_1+g_2}$.

Let $\bar\rho=\begin{pmatrix}\omega_1^{a+b+1}&*\\0&\omega_1^b\end{pmatrix}:\operatorname{I}_{\mathbb{Q}_p}\to\operatorname{GL}_2(\mathbb{F})$ be a character with…

The remainder vanishing conjecture. Under these conditions,

The Newton-polygon criterion. Then $[F,G]=0$.

Let $n=2\ell$, assume that $\operatorname{char}(F)$ is zero or sufficiently positive, and let $\mathcal{C}_m$ be the $m$-th inflectionary curve from $y^n=x^5+x^3+sx$. For…

Let $n=2\ell$, assume that $\operatorname{char}(F)$ is zero or sufficiently positive, and let $P^{\ell,*}_m$ be the atomic inflection polynomial of $y^n=x^3+\lambda x+2$. For even…

Let $n=2\ell$, assume that the characteristic of $F$ is zero or sufficiently positive, and let $P^{\ell,*}_m$ be the atomic inflection polynomial of the Weierstrass pencil…

For $a=0,\dots,3$, define the ghost series … where … with the exponents $m_n^{(a)}(l)$ prescribed by the source. Set … and let…