26 problems

Mueller–Schmidt conjecture. The logarithmic factor in their bound should be removable for all forms of degree $n\geq 3$ and, more importantly, the factor $s^2$ should be replaced b…

Let $s,d\in\mathbb{N}$, and let $F$ be an $s$-sparse polynomial whose individual degrees are bounded by $d$. A factor of $F$ is a polynomial divisor of $F$. Bisht–Volkovich factor-…

Let $f(x)$ be a univariate polynomial, and call the number of its nonzero terms its sparsity. Erdős–Rényi conjecture. A bound on the number of terms of $f(x)^2$ should imply a boun…

Let $m_1,\ldots,m_r$ be distinct primes, and let the field be the complex numbers $\mathbb{C}$. Consider Fourier characters on … A delta function is a function supported at one poi…

Maitra–Gupta–Venkateswarlu conjecture. For every $r=1,\dots,k$ and every pair $1\leq v\ne w\leq t-1$, the exponents satisfy

Algorithmic simplification conjecture. If one has prior knowledge of the degree of $f^*$ or $h^*$, the repeated-training and averaging step in Algorithm can be removed, because gen…

Kouchnirenko's conjecture. The bound $s_1\cdots s_n$ is sharp for the number of positive solutions of $C\cdot x^A=0$ when $C$ has real coefficients.

Multiplicity conjecture. The intersection multiplicity of this system at ${\mathbf 1}$ equals

Let $B_1,\ldots,B_k\subset\mathbb Z^1_{\geqslant 0}$ be support sets, let $d_i:=\min B_i>0$, and let … Assume that $j_\infty=1$, equivalently that the associated map germ is inject…

Let $p(x,y)\in\mathbb Z[x,y]$, and let $m,n$ be integers. In the relevant hyperbolic Dehn-filling applications, $p(x^m,x^n)$ is factored into irreducible factors. Lehmer's conjectu…

Let $f_{2^k,n}$ denote the sequence of values defined in the paper, and let $k\geq 1$ and $\nu\geq 0$ be integers. For each interval of integers … consider the signed sequence…

Let $f_{2,n}(z)$ be the sparse polynomial sequence, and define $\mathcal L(a_n)=a_n^2-a_{n-1}a_{n+1}$ for a sequence $(a_n)$. For positive integers $k$, let $\mathcal L^k$ denote t…

Let $f_n(z)=f_{2,n}(z)$, and for a sequence $(a_n)$ define the operator … A sequence is called $k$-log-concave when $\mathcal L^j(a_n)\geq 0$ for $j=0,1,\ldots,k$, and infinite log…

Let $p$ and $t$ be fixed, and let $f\in\mathbb{Z}[x_1]$ be an input $t$-nomial, meaning a univariate integer polynomial with at most $t$ monomial terms. Root-counting conjecture. T…

Let $F(x,y)\in\mathbb{Z}[x,y]$ be an irreducible binary form of degree $n$ with $s+1$ nonzero coefficients, and let $h$ be a positive integer. Consider the number of integer soluti…

Consider a nonzero polynomial of the form … where each $f_{ij} \in \mathbb{C}[X]$ has at most $t$ monomials. Multiplicity $\tau$-conjecture. The multiplicity of any nonzero complex…

Consider a nonzero polynomial of the form … where each $f_{ij} \in \mathbb{R}[X]$ has at most $t$ monomials. Real $\tau$-conjecture. The number of real roots of $f$ is bounded by a…

Let an extremal reciprocal primitive be an extremal reciprocal algebraic integer that is primitive, and let an octanomial be a polynomial with eight nonzero monomials. Octanomial e…

Schinzel's non-cyclotomic factorization conjecture. There are finite sets $\Omega^0\subset\mathbb{Z}^{n\times n}$ of nonsingular matrices and $\Gamma\subset\mathbb{Z}^n$ of nonzero…

Schinzel's factorization conjecture. There is a finite set of matrices $\Omega\subset\mathbb{Z}^{n\times n}$ such that, for each ${\boldsymbol{a}}\in\mathbb{Z}^n$, there are…

Factorial decay conjecture. There exists a constant $\gamma>0$ such that

Erdős–Rényi conjecture. The number of terms of $g(x)$ is bounded in terms of $l$ alone.

Polylogarithmic root-bound conjecture. There is an absolute constant $c\geq 1$ such that this trinomial has no more than $(\log p)^c$ roots in $\mathbb{F}_p$.

Brown–Dilcher–Manna conjecture. As $n\to\infty$,

For $f\in\mathbb{Z}[x_1]$, let $tau(f)$ denote its number of nonzero coefficients, and let the roots of $f$ be counted in the indicated fields. Adelic Tau conjecture. There is…