On norming systems of linear equations
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910718763204608 |
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| author | Cho, Seokjoon Conlon, David Lee, Joonkyung Skokan, Jozef Versteegen, Leo |
| author_facet | Cho, Seokjoon Conlon, David Lee, Joonkyung Skokan, Jozef Versteegen, Leo |
| contents | A system of linear equations $L$ is said to be norming if a natural functional $t_L(\cdot)$ giving a weighted count for the set of solutions to the system can be used to define a norm on the space of real-valued functions on $\mathbb{F}_q^n$ for every $n>0$. For example, Gowers uniformity norms arise in this way. In this paper, we initiate the systematic study of norming linear systems by proving a range of necessary and sufficient conditions for a system to be norming. Some highlights include an isomorphism theorem for the functional $t_L(\cdot)$, a proof that any norming system must be variable-transitive and the classification of all norming systems of rank at most two. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18389 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On norming systems of linear equations Cho, Seokjoon Conlon, David Lee, Joonkyung Skokan, Jozef Versteegen, Leo Combinatorics Number Theory A system of linear equations $L$ is said to be norming if a natural functional $t_L(\cdot)$ giving a weighted count for the set of solutions to the system can be used to define a norm on the space of real-valued functions on $\mathbb{F}_q^n$ for every $n>0$. For example, Gowers uniformity norms arise in this way. In this paper, we initiate the systematic study of norming linear systems by proving a range of necessary and sufficient conditions for a system to be norming. Some highlights include an isomorphism theorem for the functional $t_L(\cdot)$, a proof that any norming system must be variable-transitive and the classification of all norming systems of rank at most two. |
| title | On norming systems of linear equations |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2411.18389 |