6 problems

Consider a system of $n$ polynomials in $n$ variables, where the $i$th polynomial has $m_i$ monomials and all solutions are simple. Kouchnirenko's conjecture. The system has at mos…

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…

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

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…