# Applications of Supersymmetric Polynomials in Statistical Quantum Physics

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary Results for Symmetric Polynomials and Partition Functions

#### 2.1. Symmetric Polynomials

**Definition**

**1.**

#### 2.2. Partition Functions

#### 2.3. Note about the Banach Space ${\mathcal{\ell}}_{1}$

## 3. Supersymmetric Polynomials and Partition Functions for Mixed Systems of Bosons and Fermions

**Example**

**1.**

## 4. Semi-ring Structures on the Set of Variables

#### 4.1. The Ring ${\mathcal{M}}_{0}$

**Example**

**2.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

#### 4.2. A Tropical Semi-Ring Structure

**Definition**

**2.**

**Proposition**

**4.**

**Proof.**

**Theorem**

**1.**

- 1.
- The tropical operations are continuous in ${\mathcal{M}}_{X}^{\oplus}$;
- 2.
- The mappings$${\mathrm{\Phi}}^{+}\left[u\right]=\underset{i}{max}{x}_{i}\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}{\mathrm{\Phi}}^{-}\left[u\right]=\underset{j}{min}(-{y}_{j})$$

**Proof.**

## 5. Discussions and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Chernega, I.; Martsinkiv, M.; Vasylyshyn, T.; Zagorodnyuk, A.
Applications of Supersymmetric Polynomials in Statistical Quantum Physics. *Quantum Rep.* **2023**, *5*, 683-697.
https://doi.org/10.3390/quantum5040043

**AMA Style**

Chernega I, Martsinkiv M, Vasylyshyn T, Zagorodnyuk A.
Applications of Supersymmetric Polynomials in Statistical Quantum Physics. *Quantum Reports*. 2023; 5(4):683-697.
https://doi.org/10.3390/quantum5040043

**Chicago/Turabian Style**

Chernega, Iryna, Mariia Martsinkiv, Taras Vasylyshyn, and Andriy Zagorodnyuk.
2023. "Applications of Supersymmetric Polynomials in Statistical Quantum Physics" *Quantum Reports* 5, no. 4: 683-697.
https://doi.org/10.3390/quantum5040043