14 problems

Let ${\mathcal P}\subset\mathbb C^n$ be a product of discs or ${\mathcal P}\subset\mathbb R^n$ a product of intervals, and let $f_1,\ldots,f_N$ be analytic functions satisfying pol…

Let a general system consist of $m$ polynomials in $m$ variables, and measure its sparsity by the number of nonzero terms appearing among the polynomials. Kushnirenko's conjecture.…

Let a fewnomial system in $m$ variables have $|f|$ monomials in total, and consider its real zeros in the positive real quadrant. Kouchnirenko's conjecture. The maximum number of s…

Fewnomial average-case complexity conjecture. The feasibility problem $\mathbf{FEAS}_{\mathbb{R}}(\mathbb{Z}[x_1])$ is $\mathbf{NP}$-hard but, for any fixed $m$, there is a natural…

Linear-length Sturm-sequence conjecture. The maximal length of the Sturm sequence of a degree $2D$ tetranomial is

Let ${\mathcal P}$ be the domain of a Noetherian chain and let $P_1,\ldots,P_n:{\mathcal P}\to\mathbb C$ be Noetherian functions of format ${\mathscr F}$. Khovanskii's format-bound…

Bounded-chamber conjecture. Whenever the parametrization map has two critical points, the complement of the signed reduced $A$-discriminant contains a bounded connected component.

For $i=1,\ldots,n$, let $A_i\subseteq\mathbb{R}^n$ be finite nonempty sets of cardinality $t_i$, with convex hulls $P_i$. Let $V_0$ be the number of vertices of the Minkowski sum…

Let $n$ be fixed, and consider a system of $n$ real polynomial equations in $n$ variables with $t$ distinct exponent vectors. A positive solution is nondegenerate if the Jacobian o…

Polynomial-time approximate-root conjecture. Following this notation, one can find approximate roots for all the real roots of $g$ in time polynomial in $\log^n(BH)$.

Fewnomial bound optimality conjecture. These bounds are far from being optimal.

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

Let $p$ be a prime and let $A_1,\ldots,A_n\subset\mathbb{Z}^n$ be finite subsets. Write $A:=\bigcup_i A_i$ and $t:=\#A$. For a tuple $F=(f_1,\ldots,f_n)$ with…

Let $OmegaL(n,k)$ denote the maximum number of non-degenerate roots in a local field $L$ of a system of $n$ polynomials in $n$ variables having at most $k$ monomials in total. The…