$\mathcal{K}$-Lorentzian Polynomials
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | |
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| _version_ | 1866929351689240576 |
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| author | Blekherman, Grigoriy Dey, Papri |
| author_facet | Blekherman, Grigoriy Dey, Papri |
| contents | Lorentzian polynomials are a fascinating class of real polynomials with many applications. Their definition is specific to the nonnegative orthant. Following recent work, we examine Lorentzian polynomials on proper convex cones. For a self-dual cone $\mathcal{K}$ we find a connection between $\mathcal{K}$-Lorentzian polynomials and $\mathcal{K}$-positive linear maps, which were studied in the context of the generalized Perron-Frobenius theorem. We find that as the cone $\mathcal{K}$ varies, even the set of quadratic $\mathcal{K}$-Lorentzian polynomials can be difficult to understand algorithmically. We also show that, just as in the case of the nonnegative orthant, $\mathcal{K}$-Lorentzian and $\mathcal{K}$-completely log-concave polynomials coincide. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_12973 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\mathcal{K}$-Lorentzian Polynomials Blekherman, Grigoriy Dey, Papri Algebraic Geometry 14P99, 52A20 Lorentzian polynomials are a fascinating class of real polynomials with many applications. Their definition is specific to the nonnegative orthant. Following recent work, we examine Lorentzian polynomials on proper convex cones. For a self-dual cone $\mathcal{K}$ we find a connection between $\mathcal{K}$-Lorentzian polynomials and $\mathcal{K}$-positive linear maps, which were studied in the context of the generalized Perron-Frobenius theorem. We find that as the cone $\mathcal{K}$ varies, even the set of quadratic $\mathcal{K}$-Lorentzian polynomials can be difficult to understand algorithmically. We also show that, just as in the case of the nonnegative orthant, $\mathcal{K}$-Lorentzian and $\mathcal{K}$-completely log-concave polynomials coincide. |
| title | $\mathcal{K}$-Lorentzian Polynomials |
| topic | Algebraic Geometry 14P99, 52A20 |
| url | https://arxiv.org/abs/2405.12973 |