Characterization of tropical projective quadratic plane curves in terms of the eigenvalue problem

Abstract

The tropical semiring R∪{−∞} is the semiring with addition “max” and multiplication “+”. Tropical quadratic forms are represented by tropical symmetric matrices. Tropical quadratic forms with three variables define tropical projective quadratic plane curves. In this paper, we characterize tropical projective quadratic plane curves in terms of the eigenvalue problem for tropical matrices. In particular, we focus on the curved part of a tropical projective quadratic plane curve, that is, the cell never contained in any tropical projective line. We first prove that algebraic eigenvalues, i.e., roots of the characteristic polynomial, of a matrix express the minimum distance from the origin to the curved part. We then show that an algebraic eigenvector with respect to the minimum algebraic eigenvalue of a matrix indicates the direction to the nearest point in the curved part from the origin.