This following is a brief summary of our research meeting on 2024-08-02.

Summary of discoveries

• Nicholas sketched conjectured dual regions for tropical bend conics of Types $B_1$ and $E_1$. A summary of the line of thinking can be found here. In the one case, the proposed dual region has a shaded area, suggesting it should probably be a congruence variety. In the other, the dual region looked just like a tropical line. Maybe Type $G_1$ in that case?
• Aaron shared an idea of reverse engineering tropical quadratic polynomials from congruence varieties like the one in some of the proposed duals.

Questions

• Can we find a formula that will take the coefficients of the quadratic polynomial (that defines the bend conic) and produces the polynomial/equation that defines the dual region?
• There are some tropical lines that could arguably be considered tropically tangent (or not). How do we feel about those?

Tasks for next meeting

• Flesh out more examples. For conics of Type $B_1$, find equations that define the dual regions as congruence varieties.
• For the dual regions, look for some way (geometric or algebraic) that we could reasonably reconstruct the original bend conic.
• Look at some of the more ... interesting types, such as Type $E$ of $F$. Can we apply our ideas from the other types to those?
• Don't be afraid to think outside of the box!