6. Miscellaneous

Problem 6.1.
[Hector Pasten] Prove or disprove the following conjecture.
Let $K$ be a global field, let $v$ be a place of $K$, and let $X$ and $Y$ be projective varieties which are not singletons. Suppose that $X(K)$ is Zariski dense in $X$. Given a dominant rational map $f: X \dashrightarrow Y$ and a nonempty Zariski open set $U\subset X$, there is a Cartier divisor $D > 0$ on $Y$, defined over $K$, such that some sequence of points in $(U \setminus f^*D)(K)$ approaches $f^*D$ $v$adically. 
Problem 6.4.
[Rumen Dimitrov] What are the automorphisms of cohesive powers $\prod_C F$ for finite algebraic extensions $F$ of $\mathbb Q$?
 Are there any that do not arise from automorphisms of $F$?

Problem 6.5.
[Thanases Pheidas] Consider the system of equations $$ \left\{ \begin{array}{ll} a_1s_1 + \dots + a_4s_4 &= c\\ b_1s_1 + \dots + b_4s_4 &= d \end{array} \right. $$ where $a_i, b_j, c,d\in \mathbb Z$ for all $1\leq i,j\leq 4$. Suppose that this system is solvable with the $s_j$ in $\mathbb Z$ modulo every prime $p$. Fix $k \geq 4$, and let $s_j$ be the $j$th symmetric function of $x_1, \dots, x_k$. (E.g., $s_1 = x_1 + \dots + x_k$.) Must this system have integer solutions for $x_1, \dots, x_k$? What happens if we add more rows or columns?
Cite this as: AimPL: Definability and decidability problems in number theory, available at http://aimpl.org/definedecide.