# Hyperbolic and Weak Euclidean Polynomials from Wronskian and Leibniz Maps

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Remark**

**1.**

**(1)**When P, as defined above, runs through the set ${\mathbb{R}}_{n}^{e}\left[X\right]$ of Euclidean polynomials of degree n, the map $({a}_{1},\dots {a}_{n})\in {\mathbb{R}}^{n}\mapsto ({x}_{1},\dots {x}_{n})\in {\mathbb{R}}^{n}$ gives an Euclidean correspondence between the relevant subsets of ${\mathbb{R}}^{n}$.

**(2)**For any fixed pair $(n,a)\in {\mathbb{Z}}_{\ge 2}\times {\mathbb{R}}_{+}$, there is at least one Euclidean polynomial ${P}_{a}\in {\mathbb{R}}_{n}\left[X\right]$ with the given norm $E\left({P}_{a}\right)=a$. One such example is

**(3)**In [1], the following geometric characterization involving the standard unit $(n-2)$-sphere ${S}^{n-2}$ in ${\mathbb{R}}^{n-1}$ for weak Euclidean polynomials is given. The polynomial $P\in {\mathbb{R}}_{n}\left[X\right]$ is weak Euclidean if and only if

**(4)**More generally, for the polynomial P in (1), we define its Euclidean defect as

**(5)**It is well known that the characteristic polynomial of a symmetric matrix Γ with real entries is real-rooted. So, we can call Γ a (weak) Euclidean if its characteristic polynomial is so.

**(6)**The hyperbolic polynomials and their multivariate generalization (called Garding hyperbolic; see, for example, [3]) appear in a natural way in various mathematical settings from real algebraic geometry and discrete mathematics to PDEs and (polynomial) optimization. So, there is an increasing scientific interest in producing and studying some classes of hyperbolic polynomials.

## 2. The Second Degree

**Proposition**

**1.**

**Example**

**1.**

**Proposition**

**2.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Remark**

**2.**

## 3. The Third Degree

#### 3.1. The First Pair $({P}_{1},{P}_{2})$

**Proposition**

**3.**

**Example**

**6.**

**Example**

**7.**

**Proposition**

**4.**

**Example**

**8.**

#### 3.2. The Second Pair $({P}_{1},{P}_{2})$

**Proposition**

**5.**

**Example**

**9.**

**Proposition**

**6.**

**Example**

**10.**

**Example**

**11.**

## 4. The Palindromic Cubic Revisited

**Proposition**

**7.**

**Proposition**

**8.**

**Example**

**12.**

## 5. Rodrigues Sequences of Polynomials

**Definition**

**2.**

**Example**

**13.**

**Remark**

**3.**

- (I)
- Applying Proposition 1, we obtain that $W({P}_{1},{P}_{2})$ is hyperbolic if and only if$${c}^{2}-ac+{b}^{2}={\left(\frac{\alpha}{2}\right)}^{2}-\frac{{\alpha}^{2}}{2}+\frac{{\alpha}^{2}+2\beta}{6}\ge 0\to 4\beta \ge {\alpha}^{2}.$$Hence, $W({P}_{1},{P}_{2})$ and G are simultaneously hyperbolic if and only if $4\beta ={\alpha}^{2}$, i.e., $G\left(X\right)={P}_{2}\left(X\right)={\left(X+\frac{\alpha}{2}\right)}^{2}$. If $\beta ={\alpha}^{2}$, the $W({P}_{1},{P}_{2})$ is a Euclidean polynomial. $W({L}_{1}^{m},{L}_{2}^{m})$ is not hyperbolic nor Euclidean.
- (II)
- Applying Proposition 2, we have that $\frac{1}{3}L({P}_{1},{P}_{2})$ is hyperbolic if and only if$${(a+c)}^{2}-3(ac+b)=\frac{{\alpha}^{2}-4\beta}{4}\ge 0.$$Therefore, $\frac{1}{3}L({P}_{1},{P}_{2})$ is hyperbolic if and only if the generator G is hyperbolic. If $\beta =-2{\alpha}^{2}$, then $\frac{1}{3}L({P}_{1},{P}_{2})$ is a Euclidean polynomial, and $\frac{1}{3}L({L}_{1}^{m},{L}_{2}^{m})$ is a non-Euclidean hyperbolic polynomial.

## 6. Conclusions and Future Works

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Crasmareanu, M. The diagonalization map as submersion, the cubic equation as immersion and Euclidean polynomials. Mediterr. J. Math.
**2022**, 19, 65. [Google Scholar] [CrossRef] - Kostov, V.P. Topics on Hyperbolic Polynomials in One Variable; Panoramas et Synthèses: Paris, France, 2011; Volume 33. [Google Scholar]
- Rainer, A. Roots of Garding hyperbolic polynomials. Proc. Am. Math. Soc.
**2022**, 150, 2433–2446. [Google Scholar] [CrossRef] - Kurdyka, K.; Paunescu, L. Nuij type pencils of hyperbolic polynomials. Can. Math. Bull.
**2017**, 60, 561–570. [Google Scholar] [CrossRef] - Sottile, F. Real Solutions to Equations from Geometry; University Lecture Series 57; American Mathematical Society: Providence, RI, USA, 2011. [Google Scholar]
- Grechuk, B. Theorems of the 21st Century. Volume I; Springer: Cham, Switzerland, 2019. [Google Scholar]
- Weinstein, J. Reciprocity laws and Galois representations: Recent breakthroughs. Bull. Am. Math. Soc.
**2016**, 53, 1–39. [Google Scholar] [CrossRef] - Capparelli, S.; Del Fra, A.; Vietri, A. Searching for hyperbolic polynomials with span less than 4. Exp. Math.
**2022**, 31, 830–842. [Google Scholar] [CrossRef] - King, R.B. Beyond the Quartic Equation; Modern Birkhäuser Classics; Birkhäuser: Basel, Germany, 2008. [Google Scholar]
- Crasmareanu, M. Weighted Riemannian 1-manifolds for classical orthogonal polynomials and their heat kernel. Anal. Math. Phys.
**2015**, 5, 373–389. [Google Scholar] [CrossRef] - Dey, P.; Plaumann, D. Testing hyperbolicity of real polynomials. Math. Comput. Sci.
**2020**, 14, 111–121. [Google Scholar] [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Crasmareanu, M.
Hyperbolic and Weak Euclidean Polynomials from Wronskian and Leibniz Maps. *Axioms* **2024**, *13*, 104.
https://doi.org/10.3390/axioms13020104

**AMA Style**

Crasmareanu M.
Hyperbolic and Weak Euclidean Polynomials from Wronskian and Leibniz Maps. *Axioms*. 2024; 13(2):104.
https://doi.org/10.3390/axioms13020104

**Chicago/Turabian Style**

Crasmareanu, Mircea.
2024. "Hyperbolic and Weak Euclidean Polynomials from Wronskian and Leibniz Maps" *Axioms* 13, no. 2: 104.
https://doi.org/10.3390/axioms13020104