21 problems

Arnold's conjecture. The asymptotic behavior of this oscillatory integral is completely determined by the Newton polyhedron of $\phi$ in a so-called adapted coordinate system.

Let $f:\mathbb{C}^{n+1}\rightarrow\mathbb{C}$ be a convenient nondegenerate polynomial, and let the spectrum of $f$ be defined using the Newton filtration. For hypergeometric perio…

Modified Newton distance conjecture. If $S\in\mathfrak S^m\mathbb R^{n_X+n_Z}$, then

The bound conjecture. It may be conjectured that

Let $L_0(s,\xi)$ be the greatest common divisor of the polynomials $\sigma(L_1)(s,\xi),\ldots,\sigma(L_k)(s,\xi)$, and let the residual resultant be the resultant of…

Let $f$ be a singular complex analytic germ at $0\in\mathbb{C}^{n}$ with $f(0)=0$, and let $\varphi$ be a Schwarz function on $\mathbb{C}^{n}$ with sufficiently small support and…

Let $f$ be non-degenerate with respect to its Newton polyhedron, and let $\tau_{0}$ be a codimension-one simplex whose only lattice points in its intersection with the support of…

Let $f$ be a real analytic germ at $0\in\mathbb{R}^{n}$, let $\tau_{0}$ be the associated face of its Newton polyhedron, and let $\mu(\varphi)$ be the relevant coefficient of the o…

Let $f$ be a real analytic germ at $0\in\mathbb{R}^{n}$ with Newton polyhedron, and let $\tau_{0}$ be the face determining the relevant pole and coefficient $\mu(\varphi)$ of the a…

Border-splitting conjecture. Every missing facet $\delta$ in the classification can be split into two pieces $\phi$ and $\tau$ satisfying the definition of a border, with $\phi$ an…

Combinatorial triangulation conjecture.

Triangulation-dependent Łojasiewicz conjecture. Then

Brzostowski–Krasiński–Oleksik conjecture. If $\Gamma(f)$ contains a nonexceptional facet, then

Denef–Sargos conjecture. If $1/t_0\notin\mathbb{Z}$, then the pole order of $Z_f$ at $s=-1/t_0$ is equal to $\operatorname{codim}_{\mathbb{R}^n}\tau_0$ if and only if $\tau_0$ is s…

Coefficient divisibility and nonvanishing conjecture. 1. If $n=p^m$ is a power of a prime $p$, then every coefficient of $p_n$ except the coefficient of $e_1^n$ is divisible by $p$…

Let $f\in\mathbf{F}_{q}\left[x_{1},\ldots,x_{n},\left(x_{1}\ldots x_{n}\right)^{-1}\right]$ be nondegenerate and suppose that $\dim\Delta(f)=n$. Let $w_k$ be the number of roots of…

A normal surface singularity is a surface singularity that is normal, and a Newton non-degenerate singularity is one for which, for every $\mathbf{u}\in\mathbb{R}_{\geq 0}^n$, the…

Let $\Sigma_1,\ldots,\Sigma_m$ be an essential collection of integer polyhedra in $\mathbb{R}^m$, and let their mixed volume be $1/m!$, the minimal possible value. Minimal mixed-vo…

Let $F$ be a number field, let $K$ be a completion of $F$, and let $Z_{\Phi}(s,\chi,f_l,V_{l-1},K)$ be the local zeta function associated with the complex non-degenerate polynomial…

A family of hypersurface singularities is minimally obstructed when its equianalytic stratum has the minimal obstruction described in the preceding results. A hypersurface singular…

Let … be an oscillatory integral, where $f$ is its phase function and $a$ is an amplitude. A coordinate system is called adapted when the Newton distance of the phase function in t…