The idealizer of the set of quasi-stable polynomials
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910200945967104 |
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| author | Kudra, Michał |
| author_facet | Kudra, Michał |
| contents | It follows from the Garloff-Wagner Theorem that the set of stable polynomials of degree $n$, denoted by $\mathcal{H}_n$, i.e., those whose zeros all lie in the open left complex half-plane, with the Hadamard product $*$, forms an abelian semigroup contained in the abelian group $\mathbb{R}_n^+$ of polynomials of degree $n$ with positive real coefficients. By the idealizer of the set $\mathcal{H}_n$, we refer to the largest subsemigroup of $\mathbb{R}_n^+$ in which $\mathcal{H}_n$ is an ideal. In this paper, we formulate a conjecture characterizing the idealizer of $\mathcal{H}_n$ and prove it for $n \leqslant 5$. In addition, we show that the proposed condition is necessary for any polynomial to belong to the idealizer and establish, in a distinguished special case, a sufficient condition of a similar nature that supports the conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07628 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The idealizer of the set of quasi-stable polynomials Kudra, Michał Complex Variables Optimization and Control 12D10, 30C15, 93D20 It follows from the Garloff-Wagner Theorem that the set of stable polynomials of degree $n$, denoted by $\mathcal{H}_n$, i.e., those whose zeros all lie in the open left complex half-plane, with the Hadamard product $*$, forms an abelian semigroup contained in the abelian group $\mathbb{R}_n^+$ of polynomials of degree $n$ with positive real coefficients. By the idealizer of the set $\mathcal{H}_n$, we refer to the largest subsemigroup of $\mathbb{R}_n^+$ in which $\mathcal{H}_n$ is an ideal. In this paper, we formulate a conjecture characterizing the idealizer of $\mathcal{H}_n$ and prove it for $n \leqslant 5$. In addition, we show that the proposed condition is necessary for any polynomial to belong to the idealizer and establish, in a distinguished special case, a sufficient condition of a similar nature that supports the conjecture. |
| title | The idealizer of the set of quasi-stable polynomials |
| topic | Complex Variables Optimization and Control 12D10, 30C15, 93D20 |
| url | https://arxiv.org/abs/2605.07628 |