Real Algebraic and Analytic Geometry
Submission: 2007, February 13.
We introduce a family of metric invariants for tropical conics. We classify tropical conics, according to these invariants: there are proper and improper conics, and pairs of lines are a particular sort of the latter. We characterize pairs of lines in terms of the tropical singularity of certain matrix. The invariants help draw the tropical conic given by an arbitrary degree–two homogeneous tropical polynomial in three variables. It is a tree of a very particular kind. We also show how to produce a polynomial, when such a tree is given. Finally, we give criteria to decide whether a tropical degree–two homogeneous polynomial in three variables is reducible and, if so, we find a factorization.
Mathematics Subject Classification (2000): 11C99, 12K99, 05C05, 12Y05.
Keywords and Phrases: Tropical conics, metric invariants, factorization of tropical polynomials, tropically singular matrix.
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