Theory of Log-concave and Completely Log-concave Polynomials

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Moradzadeh, Ahmad Reza | 2021

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 54200 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Ebrahimi Broujeni, Javad
Abstract:
One powerful methods to study combinatorial structures is to use polynomials. From 2006 onwards, by introducing and examining the properties of a class of polynomials called stable polynomials, deep connections were made between combinatorial structures and polynomials, and this led to special applications in solving problems such as van der Waerden conjecture conjecture and the proof of existence of Ramanojan graphs. In 2018, classes of polynomials called log-concave polynomials and completely log-concave polynomials were introduced, which led to the solution of an old conjecture, called Mason conjecture, in the field of matroid theory. The main purpose of this dissertation is to study the features and structure of this category of polynomials
Keywords:
Stable Polynomials ; Completely Log-Concave Polynomials ; Log-Concave Polynomials ; Hyperbolic Polynomials ; Matroids Theory ; Mason Conjecture