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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.06440 |
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| _version_ | 1866911100951330816 |
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| author | Grigoriev, Dima |
| author_facet | Grigoriev, Dima |
| contents | For a tropical prevariety $V\subset \RR^n$ (being a finite union of rational polyhedra) we define a tropical Hilbert function $TH_V(k)$ to be the maximal number of tropically independent on $V$ among tropical monomials with degrees at most $k$. In case $\dim V=1$ we define the tropical degree as $$degT (V):=\lim_{k\to \infty} \frac{TH_V(k)}{k}$$ \noindent and prove existence of the limit. We calculate explicitly (a modification of) the tropical degree of $V$ when $V=\cup_l V_l$ is a star, i.e. the union of rays $V_l$ with a common apex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06440 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Degree of 1-dimensional tropical prevariety Grigoriev, Dima Algebraic Geometry 14T10 G.2.1 For a tropical prevariety $V\subset \RR^n$ (being a finite union of rational polyhedra) we define a tropical Hilbert function $TH_V(k)$ to be the maximal number of tropically independent on $V$ among tropical monomials with degrees at most $k$. In case $\dim V=1$ we define the tropical degree as $$degT (V):=\lim_{k\to \infty} \frac{TH_V(k)}{k}$$ \noindent and prove existence of the limit. We calculate explicitly (a modification of) the tropical degree of $V$ when $V=\cup_l V_l$ is a star, i.e. the union of rays $V_l$ with a common apex. |
| title | Degree of 1-dimensional tropical prevariety |
| topic | Algebraic Geometry 14T10 G.2.1 |
| url | https://arxiv.org/abs/2404.06440 |