Loading [MathJax]/jax/output/HTML-CSS/fonts/TeX/AMS/Regular/Main.js
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

6. Miscellaneous


    1. Add remark

      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.


        • Add remark

          Problem 6.4.

          [Rumen Dimitrov]
          1. What are the automorphisms of cohesive powers \prod_C F for finite algebraic extensions F of \mathbb Q?
          2. Are there any that do not arise from automorphisms of F?


            • Add remark

              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.