PROBLEMS

Let $\mathbb{F}_q$ be a finite field and $f(x)\in\mathbb{F}_q[x]$ nonconstant. Does there exist a deterministic polynomial-time algorithm for factoring arbitrary polynomials in $\mathbb{F}_q[x]$?

For a monomial ideal $I\subset k[x_1,\dots,x_n]$, is it always true that \[\operatorname{sdepth}(I)\ge \operatorname{depth}(I)?\]

For a reflexive lattice polytope $P\subset\mathbb{R}^d$ with \[\sum_{m\ge0} |mP\cap\mathbb{Z}^d|t^m=\frac{h^*(t)}{(1-t)^{d+1}},\] must the symmetric sequence $(h_i^*)$ always be unimodal?

Does the integer decomposition property imply unimodality or stronger positivity properties for the Ehrhart $h^*$-vector?

For a numerical semigroup $S=\langle a_1,\dots,a_e\rangle$ with Frobenius number $F$ and genus $g$, does \[F+1\le e\,g\] always hold?

If a degree $n$ polynomial $f(x)$ over characteristic $0$ satisfies $\gcd(f,f^{(i)})\neq1$ for all $1\le i\le n-1$, must $f(x)=(x-a)^n$?

If $A[t]\cong k[x_1,\dots,x_{n+1}]$ as $k$-algebras, must $A\cong k[x_1,\dots,x_n]$?

In higher dimensions, when does unirationality imply rationality for projective varieties?

For general points in $\mathbb{P}^2$ with multiplicities $m_i$, what is the minimal degree $d$ of a curve vanishing to order $m_i$ at each point?

Is every torsion unit in $\mathbb{Z}[G]$ conjugate in $\mathbb{Q}[G]$ to an element of $G$?

Can one describe $\mathrm{gin}(I)$ or its Betti numbers explicitly for ideals generated by general forms?

Which monomial ideals admit minimal free resolutions supported on CW-complexes?

Which sequences occur as the $h$-vector of a simplicial sphere?

When does a sparse binomial $x^a+cx^b\in\mathbb{F}_q[x]$ define a permutation of $\mathbb{F}_q$?

Is there a structural classification for permutation trinomials or quadrinomials over finite fields?

What Hilbert functions and Betti tables occur for almost complete intersections?

When does $I^{(m)}=I^m$ hold for an ideal $I$?

For which $m,r$ does $I^{(m)}\subseteq I^r$ hold?

Given $\mathrm{reg}(I^m)=am+b$ for $m\gg0$, how can one determine $a$ and $b$ from the generators of $I$?

For the Rees algebra $\mathcal{R}(I)=\bigoplus_{m\ge0} I^m t^m$, what are its defining equations and homological properties?