Problem 2A — What is the minimal number of open sets $U_{i}$ covering ${\mathbb{R}}^{6}$ such that for any $U_{i}$…

Definitions and mathematical statement

Let $p=(a,b)\in{\mathbb{R}}^6$ denote the source's parameter pair and let $f_a,g_b$ be its two real functions on ${\mathbb{R}}^2$. Their formulas, degrees, and parameter split are not included in the supplied record. Define the solution incidence space, conditional on that data, by \[ E=\{(p,x)\in{\mathbb{R}}^6\times{\mathbb{R}}^2:f_a(x)=0,\ g_b(x)=0\}, \qquad\pi:E\to{\mathbb{R}}^6,\quad\pi(p,x)=p. \] A continuous choice of a solution on an open set $U\subseteq{\mathbb{R}}^6$ is a map $\varphi:U\to{\mathbb{R}}^2$ with \[ f_a(\varphi(a,b))=g_b(\varphi(a,b))=0 \quad\text{for every }(a,b)\in U. \] Equivalently, $p\mapsto(p,\varphi(p))$ is a continuous section of $\pi$ over $U$.

Problem (Victor Vassiliev, Problem 2A). Determine the minimum number of open sets covering ${\mathbb{R}}^6$ on each of which such a continuous solution choice exists. If the minimum is finite, it is the least positive integer $m$ such that there are open $U_1,\ldots,U_m$ and continuous $\varphi_i:U_i\to{\mathbb{R}}^2$ satisfying \[ \bigcup_{i=1}^{m}U_i={\mathbb{R}}^6,\qquad f_a(\varphi_i(a,b))=g_b(\varphi_i(a,b))=0 \quad((a,b)\in U_i). \] The source does not remove singular parameters or require the chosen solutions to be unique, distinct, or regular. The maps need not agree on overlaps. The formulas of $f_a,g_b$ are indispensable: without them, even solvability of every fiber and existence of local sections cannot be checked. The excerpt also does not establish finiteness of the requested covering number.

Short English statement

Find the smallest open cover of the full parameter space ${\mathbb{R}}^6$ on whose members one can continuously choose a common real zero of $f_a$ and $g_b$. The defining functions are missing, so the associated solution-selection problem is not yet fully specified.

Sources

• Local UnsolvedMath registry, permanent ID 10800007, source code catalogue entry; proposer Victor Vassiliev, Problem 2A.
• Vassiliev, A Few Problems on Monodromy and Discriminants, Problem 2A. Supplied URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/; recorded access 2026-07-29; extraction html-structure. The article contains eighteen explicitly labeled theorem-style problem environments.
• Supplied status open and OPEN-TRIAGE. The research summary says a lower bound of $2$ is known and it did not determine whether this map's covering number, or higher-degree analogues, equals $2$ or is larger. The specific map definition and supporting lower-bound reference are not supplied. Current primary-literature status and redistribution terms of the publicly accessible author/publisher copy remain NEEDS_REVIEW.