187 problems

Let $A_N(\gamma)$ be the oscillatory phase average associated with the centred layer moment, let $X$ be the relevant scale, and let $R$ be the corresponding correlation quantity. S…

Let $m>1$ be an integer and let $p$ be an odd prime. Define … where $x$ denotes the fractional part of a real number $x$. Zhi-Wei Sun's conjecture. For the monomial…

Petridis–Risager's exponential sum conjecture. For every $\epsilon>0$,

Let $p$ run through the primes with $p\equiv 1\pmod 3$, and define … where $e(y)=e^{2\pi i y}$. Let … where $\Gamma$ is the classical gamma function. Patterson's conjecture. As…

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 Laurent polynomial with coefficients in $\overline{\mathbb{Q}}$, and let $\triangle$ be its Newton polytope. For each prime $p$, fix an embedding…

Main conjecture for the sliced Vinogradov system. Whenever $r,s,k\in\mathbb{N}$, $r<k$, and $\varepsilon>0$, one has

Polylogarithmic growth conjecture. There exists $k>0$ such that

Let $d\geq 3$ be an integer, let $N\in\mathbb N$, let $a_1,\ldots,a_N\in\mathbb C$, and write $\mathbb T^2$ for the two-dimensional torus. For $p\geq 2(d+1)$ and every…

Restriction conjecture. If $p\le p_{d-1}=d(d-1)$, then

Stechkin's conjecture. The quantity $\mathcal A$ is finite.

Let $phi\in\mathbb{Z}[X_1,\ldots,X_n]$ and let … be the local exponential sum, where the sum is over those $x=(x_1,\ldots,x_n)$ for which $p\mid x_j$ for every $j$. Let…

Birch's conjecture. As $p\to\infty$, the values $\operatorname{Bi}_p(a)$ become equidistributed according to the Sato–Tate measure as $a$ ranges over $\mathbb{F}_p^{\times}$.

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 $k$ be algebraically closed, let $r,s\geq 1$, and let $p$ be relatively prime to $r$ and $s$. Let … be a formal Laurent series in $k((1/t))$ with $a_{-s}\neq 0$, and let…

For $\alpha>0$, let $A(\alpha)$ be the asymptotic constant conjectured for … and let $B(\alpha)$ be the corresponding constant when $|z_k|=1$. Equality of asymptotic constants conj…

Exact unit-circle asymptotic conjecture. One can choose $B(\alpha)=\underline B(\alpha)=\overline B(\alpha)$ in Theorem $avs$. This conjecture would close the gap between the known…

Exact asymptotic formula conjecture. One can choose $A(\alpha)=\underline A(\alpha)=\overline A(\alpha)$ in Theorem $u77$. This would determine the true asymptotic constant for eve…

Vanishing conjecture. $H_k^0 = 0$ for all positive integers $k$. Consequently, $M_k(T)$ is a polynomial for all positive integers $k$.

Let $p$ be prime, let $n$ be a positive integer, and let $\mathbb{F}_{p^n}$ be the finite field with $p^n$ elements. Fix a primitive root $\vartheta$ of $\mathbb{F}_{p^n}$ and writ…

Denef–Sargos conjecture. The exponent $\sigma$ in these bounds is optimal for infinitely many $p$ and $m$.

Let $a,q\geq2$ be integers with $(a,q)=1$ and $q$ not a perfect square. Let $H,K\geq1$ and $\lambda,\mu$ be reals. Define $e(u):=e^{2\pi i u}$, let…

Let $a,q\geq2$ be integers with $(a,q)=1$ and $q$ not a perfect square, and let $H,K\geq1$ be reals. Define $e(u):=e^{2\pi i u}$, let $\left(\frac{\cdot}{\cdot}\right)$ denote the…

Let $m\geq 3$ be odd, let $n$ be a non-negative integer, and let $f$ be a quadratic polynomial in $n$ variables modulo $m$. Write $q=2\cos\left(\frac{\pi}{2m}\right)$ and let…