{"title":"分区等级和分区网格","authors":"Mohamed Omar","doi":"10.1007/s11083-024-09685-7","DOIUrl":null,"url":null,"abstract":"<p>We introduce a universal approach for applying the partition rank method, an extension of Tao’s slice rank polynomial method, to tensors that are not diagonal. This is accomplished by generalizing Naslund’s distinctness indicator to what we call a <i>partition indicator</i>. The advantages of partition indicators are two-fold: they diagonalize tensors that are constant when specified sets of variables are equal, and even in more general settings they can often substantially reduce the partition rank as compared to when a distinctness indicator is applied. The key to our discoveries is integrating the partition rank method with Möbius inversion on the lattice of partitions of a finite set. Through this we unify disparate applications of the partition rank method in the literature. We then use our theory to address a finite field analogue of a question of Erdős, thereby generalizing results of Hart and Iosevich and independently Shparlinski. Furthermore we generalize work of Pach et al. on bounding sizes of sets avoiding right triangles to bounding sizes of sets avoiding right <i>k</i>-configurations.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":"9 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Partition Rank and Partition Lattices\",\"authors\":\"Mohamed Omar\",\"doi\":\"10.1007/s11083-024-09685-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We introduce a universal approach for applying the partition rank method, an extension of Tao’s slice rank polynomial method, to tensors that are not diagonal. This is accomplished by generalizing Naslund’s distinctness indicator to what we call a <i>partition indicator</i>. The advantages of partition indicators are two-fold: they diagonalize tensors that are constant when specified sets of variables are equal, and even in more general settings they can often substantially reduce the partition rank as compared to when a distinctness indicator is applied. The key to our discoveries is integrating the partition rank method with Möbius inversion on the lattice of partitions of a finite set. Through this we unify disparate applications of the partition rank method in the literature. We then use our theory to address a finite field analogue of a question of Erdős, thereby generalizing results of Hart and Iosevich and independently Shparlinski. Furthermore we generalize work of Pach et al. on bounding sizes of sets avoiding right triangles to bounding sizes of sets avoiding right <i>k</i>-configurations.</p>\",\"PeriodicalId\":501237,\"journal\":{\"name\":\"Order\",\"volume\":\"9 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Order\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s11083-024-09685-7\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Order","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11083-024-09685-7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
我们引入了一种通用方法,用于将分区秩方法(陶氏切片秩多项式方法的扩展)应用于非对角张量。这是通过将 Naslund 的独特性指标推广到我们所说的分区指标来实现的。分区指标有两方面的优势:当指定的变量集相等时,它们能使恒定的张量对角;即使在更一般的情况下,与应用独特性指标相比,它们也能大大降低分区秩。我们发现的关键在于将分割秩方法与有限集分割晶格上的莫比乌斯反演结合起来。通过这种方法,我们统一了文献中对分区秩方法的不同应用。然后,我们用我们的理论解决了厄尔多斯问题的有限域类似问题,从而推广了哈特和伊奥塞维奇以及独立的施帕林斯基的结果。此外,我们还将 Pach 等人关于避开直角三角形的集合的边界大小的研究成果推广到避开直角 K 配置的集合的边界大小。
We introduce a universal approach for applying the partition rank method, an extension of Tao’s slice rank polynomial method, to tensors that are not diagonal. This is accomplished by generalizing Naslund’s distinctness indicator to what we call a partition indicator. The advantages of partition indicators are two-fold: they diagonalize tensors that are constant when specified sets of variables are equal, and even in more general settings they can often substantially reduce the partition rank as compared to when a distinctness indicator is applied. The key to our discoveries is integrating the partition rank method with Möbius inversion on the lattice of partitions of a finite set. Through this we unify disparate applications of the partition rank method in the literature. We then use our theory to address a finite field analogue of a question of Erdős, thereby generalizing results of Hart and Iosevich and independently Shparlinski. Furthermore we generalize work of Pach et al. on bounding sizes of sets avoiding right triangles to bounding sizes of sets avoiding right k-configurations.