# Linear tropical varieties

This page is still under heavy construction. Expect massive updates.

In light of the two leading candidates for the title of "tropical variety", we need to distinguish two competing (cooperating?) notions of tropical lines: 1) bend loci defined by linear tropical polynomials; and 2) congruence varieties defined by linear tropical equations.

# Linear bend loci

The bendy option for tropical lines is the bend locus of a linear tropical polynomial. Focusing on planar lines, these are the bend loci of polynomials of the form

(PUT SEVERAL EXAMPLES HERE)

The location of the vertex is easily deduced from the linear equation, and vice versa.

(GIVE FORMULAS HERE)

# Linear congruence varieties

The congruence option for tropical lines is the congruence variety of a quadratic linear equation, i.e., an equation of the form

(INSERT A BUNCH OF EXAMPLES)