On universal sign patterns
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909212720758784 |
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| author | Kostov, Vladimir Petrov |
| author_facet | Kostov, Vladimir Petrov |
| contents | We consider polynomials $Q:=\sum _{j=0}^da_jx^j$, $a_j\in \mathbb{R}^*$, with all roots real. When the {\em sign pattern} $σ(Q):=({\rm sgn}(a_d),{\rm sgn}(a_{d-1})$, $\ldots$, ${\rm sgn}(a_0))$ has $\tilde{c}$ sign changes, the polynomial $Q$ has $\tilde{c}$ positive and $d-\tilde{c}$ negative roots. We suppose the moduli of these roots distinct. The {\em order} of these moduli is defined when in their string as points of the positive half-axis one marks the places of the moduli of negative roots. A sign pattern $σ^0$ is {\em universal} when for any possible order of the moduli there exists a polynomial $Q$ with $σ(Q)=σ^0$ and with this order of the moduli of its roots. We show that when the polynomial $P_{m,n}:=(x-1)^m(x+1)^n$ has no vanishing coefficients, the sign pattern $σ(P_{m,n})$ is universal. We also study the question when $P_{m,n}$ can have vanishing coefficients. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_18895 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On universal sign patterns Kostov, Vladimir Petrov Classical Analysis and ODEs We consider polynomials $Q:=\sum _{j=0}^da_jx^j$, $a_j\in \mathbb{R}^*$, with all roots real. When the {\em sign pattern} $σ(Q):=({\rm sgn}(a_d),{\rm sgn}(a_{d-1})$, $\ldots$, ${\rm sgn}(a_0))$ has $\tilde{c}$ sign changes, the polynomial $Q$ has $\tilde{c}$ positive and $d-\tilde{c}$ negative roots. We suppose the moduli of these roots distinct. The {\em order} of these moduli is defined when in their string as points of the positive half-axis one marks the places of the moduli of negative roots. A sign pattern $σ^0$ is {\em universal} when for any possible order of the moduli there exists a polynomial $Q$ with $σ(Q)=σ^0$ and with this order of the moduli of its roots. We show that when the polynomial $P_{m,n}:=(x-1)^m(x+1)^n$ has no vanishing coefficients, the sign pattern $σ(P_{m,n})$ is universal. We also study the question when $P_{m,n}$ can have vanishing coefficients. |
| title | On universal sign patterns |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2405.18895 |