The Solution Of Tropical Polynomials And The Research Of Rank Of Tropical Matrices

Tropical geometry,an important branch of algebraic geometry,is based on tropical semiring,which is the set Rmax:=R∪{-∞} with two kinds of operations addition ⊕ and multiplication?,where the(Rmax,⊕)form a monoid.To address the non-invertibility of addition in the tropical semiring,M.Plus introduced the symmetric max-plus semiring Smax,which has enriched the field of tropical geometry.The applications of these semirings extend beyond tropical geometry,encompassing fields such as computer science and operations research.Polynomials and matrices are fundamental objects in algebra.However,there has been limited research on tropical polynomials and matrices.In this work,we aim to initiate research on the root of tropical polynomials and the rank of matrices.In contrast to polynomials over fields,tropical polynomials are piecewise linear convex functions.The concept of rank takes on various definitions in tropical matrices,and has therefore received significant attention.This thesis aims to investigate the existence and solution formula of tropical polynomials over a single variable on the symmetric max-plus semiring Smax,as well as explore the relationship between quasi-singularity of tropical matrices and elementary matrices.Additionally,we study the weakly row(column)rank of matrices on Smax by analyzing the linear dependence of its row(column)vectors.Notably,our results reveal the existence of 4 linearly independent 3-dimensional vectors.