{"title":"有限域上的不可满足方程组","authors":"Alan R. Woods","doi":"10.1109/SFCS.1998.743444","DOIUrl":null,"url":null,"abstract":"The properties of any system of k simultaneous equations in n variables over GF(q), are studied, with a particular emphasis on unsatisfiable systems. A general formula for the number of solutions is given, which can actually be useful for computing that number in the special case where all the equations are of degree 2. When such a quadratic system has no solution, there is always a proof of unsatisfiability of size q/sup n/2/ times a polynomial in n and q, which can be checked deterministically in time satisfying a similar bound. Such a proof can be found by a probabilistic algorithm in time asymptotic to that required to test, by substitution in k quadratic equations, all q/sup n/ potential solutions.","PeriodicalId":228145,"journal":{"name":"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)","volume":"41 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1998-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"Unsatisfiable systems of equations, over a finite field\",\"authors\":\"Alan R. Woods\",\"doi\":\"10.1109/SFCS.1998.743444\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The properties of any system of k simultaneous equations in n variables over GF(q), are studied, with a particular emphasis on unsatisfiable systems. A general formula for the number of solutions is given, which can actually be useful for computing that number in the special case where all the equations are of degree 2. When such a quadratic system has no solution, there is always a proof of unsatisfiability of size q/sup n/2/ times a polynomial in n and q, which can be checked deterministically in time satisfying a similar bound. Such a proof can be found by a probabilistic algorithm in time asymptotic to that required to test, by substitution in k quadratic equations, all q/sup n/ potential solutions.\",\"PeriodicalId\":228145,\"journal\":{\"name\":\"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)\",\"volume\":\"41 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1998-11-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/SFCS.1998.743444\",\"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 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/SFCS.1998.743444","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Unsatisfiable systems of equations, over a finite field
The properties of any system of k simultaneous equations in n variables over GF(q), are studied, with a particular emphasis on unsatisfiable systems. A general formula for the number of solutions is given, which can actually be useful for computing that number in the special case where all the equations are of degree 2. When such a quadratic system has no solution, there is always a proof of unsatisfiability of size q/sup n/2/ times a polynomial in n and q, which can be checked deterministically in time satisfying a similar bound. Such a proof can be found by a probabilistic algorithm in time asymptotic to that required to test, by substitution in k quadratic equations, all q/sup n/ potential solutions.