Every real-rooted exponential polynomial is the restriction of a Lee-Yang polynomial
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| Format: | Preprint |
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2023
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| _version_ | 1866913535776260096 |
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| author | Alon, Lior Cohen, Alex Vinzant, Cynthia |
| author_facet | Alon, Lior Cohen, Alex Vinzant, Cynthia |
| contents | A Lee-Yang polynomial $ p(z_{1},\ldots,z_{n}) $ is a polynomial that has no zeros in the polydisc $ \mathbb{D}^{n} $ and its inverse $ (\mathbb{C}\setminus\overline{\mathbb{D}})^{n} $. We show that any real-rooted exponential polynomial of the form $f(x) = \sum_{j=0}^s c_j e^{λ_j x}$ can be written as the restriction of a Lee-Yang polynomial to a positive line in the torus. Together with previous work by Olevskii and Ulanovskii, this implies that the Kurasov-Sarnak construction of $ \mathbb{N} $-valued Fourier quasicrystals from stable polynomials comprises every possible $ \mathbb{N} $-valued Fourier quasicrystal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_03201 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Every real-rooted exponential polynomial is the restriction of a Lee-Yang polynomial Alon, Lior Cohen, Alex Vinzant, Cynthia Complex Variables Mathematical Physics 45B10, 52C23 A Lee-Yang polynomial $ p(z_{1},\ldots,z_{n}) $ is a polynomial that has no zeros in the polydisc $ \mathbb{D}^{n} $ and its inverse $ (\mathbb{C}\setminus\overline{\mathbb{D}})^{n} $. We show that any real-rooted exponential polynomial of the form $f(x) = \sum_{j=0}^s c_j e^{λ_j x}$ can be written as the restriction of a Lee-Yang polynomial to a positive line in the torus. Together with previous work by Olevskii and Ulanovskii, this implies that the Kurasov-Sarnak construction of $ \mathbb{N} $-valued Fourier quasicrystals from stable polynomials comprises every possible $ \mathbb{N} $-valued Fourier quasicrystal. |
| title | Every real-rooted exponential polynomial is the restriction of a Lee-Yang polynomial |
| topic | Complex Variables Mathematical Physics 45B10, 52C23 |
| url | https://arxiv.org/abs/2303.03201 |