Tropski polkolobar in simetrični Barvinokov rang : delo diplomskega seminarja

Štebljaj, Zala (author), Dolžan, David (mentor)

Najprej se seznanimo s tropskim polkolobarjem, polinomi nad tropskim polkolobarjem in tropskimi matrikami. Glavna tema je simetrični Barvinokov rang, eden izmed mnogih možnih rangov tropskih matrik. Preučimo matrike z rangom $1$ in tiste z neskončnim rangom ter si ogledamo transformacijo matrik, ki ohranja rang. Poiščemo tudi zgornjo mejo za simetrični Barvinokov rang, pri čemer ugotovimo, da rang lahko preseže velikost matrike. Pri obravnavi ranga si pomagamo tudi s teorijo grafov in na ta način zgornjo mejo še nekoliko izboljšamo. Teorijo grafov uporabimo tudi pri obravnavi primera matrike z rangom, večjim od njene velikosti.

• Language: Slovenian
• Keywords: tropski polkolobar, simetrična tropska matrika, Barvinokov rang, graf
• Work type: Bachelor thesis/paper
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2022
• PID: 20.500.12556/RUL-139029
• UDC: 512
• COBISS.SI-ID: 119541507
• Publication date in RUL: 29.08.2022

Secondary language

• Language: English
• Title: Tropical semiring and symmetric Barvinok rank

Abstract

First we get familiar with the tropical semiring, polynomials over tropical semiring and tropical matrices. The main theme is the symmetric Barvinok rank, one of the many possible ranks of tropical matrices. We examine matrices with rank $1$ and those with infinite rank and look at the transformation of matrices that preserve rank. We also find the upper bound for the symmetric Barvinok rank, noting that the rank can exceed the size of the matrix. When dealing with rank, we also use graph theory to improve the upper limit a bit. Graph theory is also used to consider an example of a matrix with a rank greater than its size.

• Keywords: tropical semiring, symmetric tropical matrix, Barvinok rank, graph