The tropical non-properness set of a polynomial map
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913435816558592 |
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| author | Hilany, Boulos El |
| author_facet | Hilany, Boulos El |
| contents | We study some discrete invariants of Newton non-degenerate polynomial maps $f : \mathbb{K}^n \to \mathbb{K}^n$ defined over an algebraically closed field of Puiseux series $\mathbb{K}$, equipped with a non-trivial valuation. It is known that the set $\mathcal{S}(f)$ of points at which $f$ is not finite forms an algebraic hypersurface in $\mathbb{K}^n$. The coordinate-wise valuation of $\mathcal{S}(f)\cap (\mathbb{K}^*)^n$ is a piecewise-linear object in $\mathbb{R}^n$, which we call the tropical non-properness set of $f$. We show that the tropical polynomial map corresponding to $f$ has fibers satisfying a particular combinatorial degeneracy condition exactly over points in the tropical non-properness set of $f$. We then use this description to outline a polyhedral method for computing this set, and to recover the fan dual to the Newton polytope of the set at which a complex polynomial map is not finite. The proofs rely on classical correspondence and structural results from tropical geometry, combined with a new description of $\mathcal{S}(f)$ in terms of multivariate resultants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_00989 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The tropical non-properness set of a polynomial map Hilany, Boulos El Algebraic Geometry Combinatorics Primary 14D06, Secondary: 14T20, 14M25, 58K15 We study some discrete invariants of Newton non-degenerate polynomial maps $f : \mathbb{K}^n \to \mathbb{K}^n$ defined over an algebraically closed field of Puiseux series $\mathbb{K}$, equipped with a non-trivial valuation. It is known that the set $\mathcal{S}(f)$ of points at which $f$ is not finite forms an algebraic hypersurface in $\mathbb{K}^n$. The coordinate-wise valuation of $\mathcal{S}(f)\cap (\mathbb{K}^*)^n$ is a piecewise-linear object in $\mathbb{R}^n$, which we call the tropical non-properness set of $f$. We show that the tropical polynomial map corresponding to $f$ has fibers satisfying a particular combinatorial degeneracy condition exactly over points in the tropical non-properness set of $f$. We then use this description to outline a polyhedral method for computing this set, and to recover the fan dual to the Newton polytope of the set at which a complex polynomial map is not finite. The proofs rely on classical correspondence and structural results from tropical geometry, combined with a new description of $\mathcal{S}(f)$ in terms of multivariate resultants. |
| title | The tropical non-properness set of a polynomial map |
| topic | Algebraic Geometry Combinatorics Primary 14D06, Secondary: 14T20, 14M25, 58K15 |
| url | https://arxiv.org/abs/2207.00989 |