{"title":"在尺寸均匀性和深度四种公式上具有较低的个别化程度","authors":"N. Kayal, Chandan Saha, Sébastien Tavenas","doi":"10.1145/2897518.2897550","DOIUrl":null,"url":null,"abstract":"Let r be an integer. Let us call a polynomial f as a multi-r-ic polynomial if the degree of f with respect to any variable is at most r (this generalizes the notion of multilinear polynomials). We investigate arithmetic circuits in which the output is syntactically forced to be a multi-r-ic polynomial and refer to these as multi-r-ic circuits. Specifically, first define the formal degree of a node a with respect to a variable x inductively as follows. For a leaf it is 1 if a is labelled with x and zero otherwise; for an internal node labelled with * (respectively +) it is the sum of (respectively the maximum of) the formal degrees of the children with respect to x. We call an arithmetic circuit as a multi-r-ic circuit if the formal degree of the output node with respect to any variable is at most r. We prove lower bounds for various subclasses of multi-r-ic circuits.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"25 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2016-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"27","resultStr":"{\"title\":\"On the size of homogeneous and of depth four formulas with low individual degree\",\"authors\":\"N. Kayal, Chandan Saha, Sébastien Tavenas\",\"doi\":\"10.1145/2897518.2897550\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let r be an integer. Let us call a polynomial f as a multi-r-ic polynomial if the degree of f with respect to any variable is at most r (this generalizes the notion of multilinear polynomials). We investigate arithmetic circuits in which the output is syntactically forced to be a multi-r-ic polynomial and refer to these as multi-r-ic circuits. Specifically, first define the formal degree of a node a with respect to a variable x inductively as follows. For a leaf it is 1 if a is labelled with x and zero otherwise; for an internal node labelled with * (respectively +) it is the sum of (respectively the maximum of) the formal degrees of the children with respect to x. We call an arithmetic circuit as a multi-r-ic circuit if the formal degree of the output node with respect to any variable is at most r. We prove lower bounds for various subclasses of multi-r-ic circuits.\",\"PeriodicalId\":442965,\"journal\":{\"name\":\"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing\",\"volume\":\"25 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2016-06-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"27\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/2897518.2897550\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/2897518.2897550","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the size of homogeneous and of depth four formulas with low individual degree
Let r be an integer. Let us call a polynomial f as a multi-r-ic polynomial if the degree of f with respect to any variable is at most r (this generalizes the notion of multilinear polynomials). We investigate arithmetic circuits in which the output is syntactically forced to be a multi-r-ic polynomial and refer to these as multi-r-ic circuits. Specifically, first define the formal degree of a node a with respect to a variable x inductively as follows. For a leaf it is 1 if a is labelled with x and zero otherwise; for an internal node labelled with * (respectively +) it is the sum of (respectively the maximum of) the formal degrees of the children with respect to x. We call an arithmetic circuit as a multi-r-ic circuit if the formal degree of the output node with respect to any variable is at most r. We prove lower bounds for various subclasses of multi-r-ic circuits.