Word maps, polynomial maps and image ratios
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909211278966784 |
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| author | Panja, Saikat |
| author_facet | Panja, Saikat |
| contents | If $A$ is a finite group (or a finite ring) and $ω$ is a word map (or a polynomial map), we define the quantity $|ω(A)|/|A|$ as the image ratio of $ω$ on $A$ and will be denoted by $μ(ω,A)$. In this article, we investigate the set $\mathrm{R}(ω)=\{μ(ω,A) : A \text{ is a finite group}\}$, and also consider the case of rings. Specifically, we demonstrate the existence of word maps (and polynomial maps) whose set of image ratios is dense in $[0,1]$ for both groups (and rings). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17026 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Word maps, polynomial maps and image ratios Panja, Saikat Group Theory Rings and Algebras 20G40, 20P05, 16R10, 16S50 If $A$ is a finite group (or a finite ring) and $ω$ is a word map (or a polynomial map), we define the quantity $|ω(A)|/|A|$ as the image ratio of $ω$ on $A$ and will be denoted by $μ(ω,A)$. In this article, we investigate the set $\mathrm{R}(ω)=\{μ(ω,A) : A \text{ is a finite group}\}$, and also consider the case of rings. Specifically, we demonstrate the existence of word maps (and polynomial maps) whose set of image ratios is dense in $[0,1]$ for both groups (and rings). |
| title | Word maps, polynomial maps and image ratios |
| topic | Group Theory Rings and Algebras 20G40, 20P05, 16R10, 16S50 |
| url | https://arxiv.org/abs/2405.17026 |