Real Algebraic and Analytic Geometry |

Metric invariants of tropical conics and factorization of degree–two homogeneous polynomials in three variables.

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Submission: 2007, February 13.

*Abstract:
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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