# REU Meeting - 2024-08-02

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
and . 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 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
, 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
of . Can we apply our ideas from the other types to those? - Don't be afraid to think outside of the box!