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| Format: | Preprint |
| Published: |
2022
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| Online Access: | https://arxiv.org/abs/2209.13811 |
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| _version_ | 1866910337032257536 |
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| author | Bisain, Ankit |
| author_facet | Bisain, Ankit |
| contents | For a complex polynomial $P$ of degree $n$ and an $m$-tuple of distinct complex numbers $Λ=(λ_1,\ldots,λ_m)$, the dope matrix $D_P(Λ)$ is defined as the $m \times (n+1)$ matrix $(c)_{ij}$ with $c_{ij} =1$ if $P^{(j)}(λ_i)=0$ and $c_{ij}=0$ otherwise. We classify the set of dope matrices when the entries of $Λ$ are algebraically independent, resolving a conjecture of Alon, Kravitz, and O'Bryant. We also provide asymptotic upper and lower bounds on the total number of $m \times (n+1)$ dope matrices. For $m$ much smaller than $n$, these bounds give an asymptotic estimate of the logarithm of the number of $m \times (n+1)$ dope matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_13811 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Generic Classification and Asymptotic Enumeration of Dope Matrices Bisain, Ankit Combinatorics For a complex polynomial $P$ of degree $n$ and an $m$-tuple of distinct complex numbers $Λ=(λ_1,\ldots,λ_m)$, the dope matrix $D_P(Λ)$ is defined as the $m \times (n+1)$ matrix $(c)_{ij}$ with $c_{ij} =1$ if $P^{(j)}(λ_i)=0$ and $c_{ij}=0$ otherwise. We classify the set of dope matrices when the entries of $Λ$ are algebraically independent, resolving a conjecture of Alon, Kravitz, and O'Bryant. We also provide asymptotic upper and lower bounds on the total number of $m \times (n+1)$ dope matrices. For $m$ much smaller than $n$, these bounds give an asymptotic estimate of the logarithm of the number of $m \times (n+1)$ dope matrices. |
| title | Generic Classification and Asymptotic Enumeration of Dope Matrices |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2209.13811 |