# Supersymmetric Polynomials on the Space of Absolutely Convergent Series

^{*}

## Abstract

**:**

## 1. Introduction and Preliminaries

## 2. Results

#### 2.1. Bases of Supersymmetric Polynomials

**Definition**

**1.**

**Theorem**

**1.**

- 1.
- ${T}_{k}(z\u2022w)={T}_{k}\left(z\right)+{T}_{k}\left(w\right)$ for every $k\in \mathbb{N}$.
- 2.
- The operations in (6) are well defined, that is, they do not depend on the choice of representatives.
- 3.
- $(\mathcal{M},\u2022,\left[z\right]\mapsto {\left[z\right]}^{-})$ is a commutative group with zero $0=\left(0\right|0)$.
- 4.
- $z\sim 0$ if and only if there are $d,s\in {\ell}_{1}$ such that $z=\left(d\right|s)$ and ${F}_{k}\left(d\right)={F}_{k}\left(s\right)$ for all $k\in \mathbb{N}$. Equivalently, all nonzero coordinates of d coincides with nonzero coordinates of s up to a permutation.

**Proof.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Corollary**

**3.**

**Corollary**

**4.**

**Proposition**

**4.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### 2.2. The Spectrum of ${H}_{B}^{Sup}$ and the Nonlinear Normed Ring $\mathcal{M}$

#### 2.2.1. The Spectrum

#### 2.2.2. The Normed Ring Structure of $\mathcal{M}$

**Proposition**

**5.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Definition**

**2.**

**Proposition**

**6.**

- 1.
- $\parallel u\parallel \ge 0$ and $\parallel u\parallel =0$ if and only if $u=0.$
- 2.
- $\parallel \lambda u\parallel =\left|\lambda \right|\parallel u\parallel .$
- 3.
- $\parallel u\u2022v\parallel \le \parallel u\parallel +\parallel v\parallel .$
- 4.
- $\parallel u\diamond v\parallel \le \parallel u\parallel \parallel v\parallel .$
- 5.
- $\parallel {u}^{-}\parallel =\parallel u\parallel .$
- 6.
- $\parallel u\parallel =\underset{\left(y\right|x)\in u}{min}(\parallel x\parallel +\parallel y\parallel ).$

**Proof.**

**Proposition**

**7.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Proposition**

**8.**

**Proof.**

**Theorem**

**6.**

**Proof.**

#### 2.2.3. Invertibility and Homomorphisms

**Proposition**

**9.**

**Proof.**

**Proposition**

**10.**

**Proof.**

**Example**

**1.**

- 1.
- Polynomials ${T}_{n},$ $n\in \mathbb{N}$ are (continuous) complex valued ring homomorphism of $\mathcal{M}$ but only ${T}_{1}$ preserves the multiplication by constants.
- 2.
- Let $u=\left[\right(y\left|x\right)]\in \mathcal{M}.$ We define$$\Theta \left(u\right)=\sum _{n=1}^{\infty}|{x}_{n}|-\sum _{n=1}^{\infty}|{y}_{n}|.$$Clearly, Θ is well defined. The additivity and multiplicativity will be proved for more general case.

**Example**

**2.**

**Proposition**

**11.**

**Proof.**

**Example**

**3.**

- 1.
- Let g be a multiplicative function from $\mathbb{N}\to \mathbb{C}.$ In Number Theory such functions are called completely multiplicative arithmetic functions. Then for $\gamma =\left|g\right|,$ ${\Theta}_{\gamma}$ is a complex valued ring homomorphisms of ${\mathcal{M}}_{S}$ and ${\mathcal{M}}_{1}.$
- 2.
- Let $\epsilon <1$ and $\epsilon \Delta $ be the closed disk in $\mathbb{C}$ of radius $\epsilon ,$ centered at zero. Then ${\mathcal{M}}_{\epsilon \Delta}$ is an ideal in ${\mathcal{M}}_{1}.$ Let$${\chi}_{\mathbb{C}\backslash \epsilon \Delta}\left(t\right)=\left\{\begin{array}{cc}0\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}\left|t\right|\le \epsilon \hfill \\ 1\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}\left|t\right|>\epsilon ,\hfill \end{array}\right.$$

#### 2.2.4. Additive Operator Calculus

**Proposition**

**12.**

**Proof.**

**Example**

**4.**

**Theorem**

**7.**

**Proof.**

**Proposition**

**13.**

- 1.
- The operator ${A}_{v}$ is bijective if and only if v is invertible in $\mathcal{M}.$
- 2.
- If the operator ${A}_{v}$ is surjective, then it is bijective.
- 3.
- The operator ${A}_{v}$ is injective if and only if $ker{A}_{v}=0.$
- 4.
- If $u\in ker{A}_{v}$ for some $u\ne 0,$ then ${T}_{n}\left(v\right)=0$ for some $n\in \mathbb{N}.$
- 5.
- If ${T}_{n}\left(v\right)=0$ for some $n\in \mathbb{N},$ then ${A}_{v}$ is not surjective.

**Proof.**

## 3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Jawad, F.; Zagorodnyuk, A.
Supersymmetric Polynomials on the Space of Absolutely Convergent Series. *Symmetry* **2019**, *11*, 1111.
https://doi.org/10.3390/sym11091111

**AMA Style**

Jawad F, Zagorodnyuk A.
Supersymmetric Polynomials on the Space of Absolutely Convergent Series. *Symmetry*. 2019; 11(9):1111.
https://doi.org/10.3390/sym11091111

**Chicago/Turabian Style**

Jawad, Farah, and Andriy Zagorodnyuk.
2019. "Supersymmetric Polynomials on the Space of Absolutely Convergent Series" *Symmetry* 11, no. 9: 1111.
https://doi.org/10.3390/sym11091111