Factorization of polynomials in one variable over the tropical semiring

2005/01/11 by Ki Hang Kim, Kim, Ki Hang, Fred W. Roush +1
Mathematics · #16Y60 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:16Y60

paper · pdf · doi:10.48550/arxiv.math/0501167

Prop.11 now assumes matrices are two-valued

arxiv created 2005/04/28 · arxiv updated 2009/12/01

Abstract

We show factorization of polynomials in one variable over the tropical semiring is in general NP-complete, either if all coefficients are finite, or if all are either 0 or infinity (Boolean case). We give algorithms for the factorization problem which are not polynomial time in the degree, but are polynomial time for polynomials of fixed degree. For two-variable polynomials we derive an irreducibility criterion which is almost always satisfied, even for fixed degree, and is polynomial time in the degree. We prove there are unique least common multiples of tropical polynomials, but not unique greatest common divisors. We show that if two polynomials in one variable have a common tropical factor, then their eliminant matrix is singular in the tropical sense. We prove the problem of determining tropical rank is NP-hard.

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• Factorizations of tropical and sign polynomials
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