This paper deals with the two following topics: bounds for the heights of divisors of polynomials, minimal distance between distinct roots of integral univariate polynomials. In each case we recall the best known results, we give some new inequalities and, constructing suitable examples, we show that these inequalities are not “too bad”.
{"title":"Some inequalities about univariate polynomials","authors":"M. Mignotte","doi":"10.1145/800206.806395","DOIUrl":"https://doi.org/10.1145/800206.806395","url":null,"abstract":"This paper deals with the two following topics: bounds for the heights of divisors of polynomials, minimal distance between distinct roots of integral univariate polynomials. In each case we recall the best known results, we give some new inequalities and, constructing suitable examples, we show that these inequalities are not “too bad”.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123145808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we present a general description of a computationally efficient algorithm for constructing every n-dimensional nilpotent Lie algebra as a central extension of a nilpotent Lie algebra of dimension less than n. As an application of the algorithm, we present a complete list of all real nilpotent six-dimensional Lie algebras. Since 1958, four such lists have been developed: namely, those of Morozov [2], Shedler [3], Vergne [5] and Skjelbred and Sund [4]. No two of these lists agree exactly. Our list resolves all the discrepancies in the other four lists. Moreover, it contains each earlier list as a subset.
本文给出了构造一个n维幂零李代数作为维数小于n的幂零李代数的中心扩展的计算效率算法的一般描述。作为该算法的一个应用，我们给出了所有实数幂零六维李代数的完整列表。自1958年以来，Morozov[2]、Shedler[3]、Vergne[5]、Skjelbred and Sund[4]等四种名单相继问世。没有哪两个列表完全一致。我们的清单解决了其他四个清单中的所有差异。此外，它将每个早期列表作为子集包含。
{"title":"Construction of nilpotent Lie algebras over arbitrary fields","authors":"R. Beck, B. Kolman","doi":"10.1145/800206.806390","DOIUrl":"https://doi.org/10.1145/800206.806390","url":null,"abstract":"In this paper we present a general description of a computationally efficient algorithm for constructing every n-dimensional nilpotent Lie algebra as a central extension of a nilpotent Lie algebra of dimension less than n.\u0000 As an application of the algorithm, we present a complete list of all real nilpotent six-dimensional Lie algebras. Since 1958, four such lists have been developed: namely, those of Morozov [2], Shedler [3], Vergne [5] and Skjelbred and Sund [4]. No two of these lists agree exactly. Our list resolves all the discrepancies in the other four lists. Moreover, it contains each earlier list as a subset.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125080076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper describes the construction of a rational function package where the GCD and factorization routines are well integrated and consistent with each other and both use state of the art algorithms. The work represents an exercise in producing a service rather than an experimental piece of code, where portability, reliability and clear readable code are important aims in addition to the obvious desire for speed. Measurements on the initial version of our package showed that even though it was based on the best of previously published methods its performance was uneven. The causes of the more notable bottle necks and the steps we took to avoid them are explained here and illustrate how apparently very fine details of coding can sometimes have gross effects on a system's overall behaviour.
{"title":"Implementing a polynomial factorization and GCD package","authors":"P. Moore, A. Norman","doi":"10.1145/800206.806379","DOIUrl":"https://doi.org/10.1145/800206.806379","url":null,"abstract":"This paper describes the construction of a rational function package where the GCD and factorization routines are well integrated and consistent with each other and both use state of the art algorithms. The work represents an exercise in producing a service rather than an experimental piece of code, where portability, reliability and clear readable code are important aims in addition to the obvious desire for speed. Measurements on the initial version of our package showed that even though it was based on the best of previously published methods its performance was uneven. The causes of the more notable bottle necks and the steps we took to avoid them are explained here and illustrate how apparently very fine details of coding can sometimes have gross effects on a system's overall behaviour.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116020887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The complicated coordinate transformations in general relativity make coordinate invariant classification schemes extremely important. A computer program, written in SHEEP, performing an algorithmic classification of the curvature tensor and a number of its derivatives is presented. The output is a complete description of the geometry. The problem to decide whether or not two solutions of Einstein's equations describe the same gravitational field can be solved if the (non-) existence of a solution to a set of algebraic equations can be established. The classification procedure has been carried through for a number of fields, and solutions previously believed to describe physically different situations have been shown to be equivalent. We exemplify with a physically interesting class of geometries.
{"title":"An algorithmic classification of geometries in general relativity","authors":"J. E. Åman, A. Karlhede","doi":"10.1145/800206.806374","DOIUrl":"https://doi.org/10.1145/800206.806374","url":null,"abstract":"The complicated coordinate transformations in general relativity make coordinate invariant classification schemes extremely important. A computer program, written in SHEEP, performing an algorithmic classification of the curvature tensor and a number of its derivatives is presented. The output is a complete description of the geometry. The problem to decide whether or not two solutions of Einstein's equations describe the same gravitational field can be solved if the (non-) existence of a solution to a set of algebraic equations can be established. The classification procedure has been carried through for a number of fields, and solutions previously believed to describe physically different situations have been shown to be equivalent. We exemplify with a physically interesting class of geometries.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127292311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Most work on computer programs to find closed form solutions to ordinary differential equations (o.d.e.s) has concentrated on implementing a catalog of those methods often cited in textbooks and reference works (see e.g. Kam61a, Inc44a]): algorithms of certain, easily recognized cases (e.g. separable, exact, homogeneous equations) and a useful guessing framework for changes of variable. This approach has been followed by Moses, Schmidt, and Lafferty, among others [Mos67a, Sch76a], [Sch79a, Laf80a]. We present here a different approach to cataloguing, using the relation between differential equations and Lie transformation groups. When presented with a given a first order o.d.e., we shall be concerned with finding continuous transformations (of the plane) which map the solution curves of the o.d.e. into each other. When a group of such transformations is found, it is possible to construct the solution to the o.d.e. via quadratures. We shall find that for many cases of interest, there are succinct algorithms for finding the transformations without knowing the solution curves beforehand. The guiding relationships for the catalogue search, and the justification for the quadrature formula used, has been known for about a century. Pioneering work was done by Sophus Lie and others in the 19th century (see e.g. [Lie75a]). Its emplacement within a symbolic/algebraic computational setting is, of course, modern-day.
{"title":"Using Lie transformation groups to find closed form solutions to first order ordinary differential equations","authors":"B. Char","doi":"10.1145/800206.806370","DOIUrl":"https://doi.org/10.1145/800206.806370","url":null,"abstract":"Most work on computer programs to find closed form solutions to ordinary differential equations (o.d.e.s) has concentrated on implementing a catalog of those methods often cited in textbooks and reference works (see e.g. Kam61a, Inc44a]): algorithms of certain, easily recognized cases (e.g. separable, exact, homogeneous equations) and a useful guessing framework for changes of variable. This approach has been followed by Moses, Schmidt, and Lafferty, among others [Mos67a, Sch76a], [Sch79a, Laf80a].\u0000 We present here a different approach to cataloguing, using the relation between differential equations and Lie transformation groups. When presented with a given a first order o.d.e., we shall be concerned with finding continuous transformations (of the plane) which map the solution curves of the o.d.e. into each other. When a group of such transformations is found, it is possible to construct the solution to the o.d.e. via quadratures. We shall find that for many cases of interest, there are succinct algorithms for finding the transformations without knowing the solution curves beforehand. The guiding relationships for the catalogue search, and the justification for the quadrature formula used, has been known for about a century. Pioneering work was done by Sophus Lie and others in the 19th century (see e.g. [Lie75a]). Its emplacement within a symbolic/algebraic computational setting is, of course, modern-day.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"58 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134476772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The determination of solutions of a system of algebraic equations is still a problem for which an efficient solution does not exist. In the last few years several authors have suggested new or refined methods, but none of them seems to be satisfactory. In this paper we are mainly concerned with exploring the use of Buchberger's algorithm for finding Groebner ideal bases [2] and combine/compare it with the more familiar methods of polynomial remainder sequences (pseudo-division) and of variable elimination (resultants) [4].
{"title":"On solving systems of algebraic equations via ideal bases and elimination theory","authors":"M. Pohst, D. Yun","doi":"10.1145/800206.806397","DOIUrl":"https://doi.org/10.1145/800206.806397","url":null,"abstract":"The determination of solutions of a system of algebraic equations is still a problem for which an efficient solution does not exist. In the last few years several authors have suggested new or refined methods, but none of them seems to be satisfactory. In this paper we are mainly concerned with exploring the use of Buchberger's algorithm for finding Groebner ideal bases [2] and combine/compare it with the more familiar methods of polynomial remainder sequences (pseudo-division) and of variable elimination (resultants) [4].","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123825418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is not always possible and sometimes not even advantageous to write the solutions of a system of differential equations explicitly in terms of elementary functions. Sometimes, though, it is possible to find elementary functions which are constant on solution curves, that is, elementary first integrals. These first integrals allow one to occasionally deduce properties that an explicit solution would not necessarily reveal.
{"title":"Elementary first integrals of differential equations","authors":"M. J. Prelle, M. Singer","doi":"10.1145/800206.806368","DOIUrl":"https://doi.org/10.1145/800206.806368","url":null,"abstract":"It is not always possible and sometimes not even advantageous to write the solutions of a system of differential equations explicitly in terms of elementary functions. Sometimes, though, it is possible to find elementary functions which are constant on solution curves, that is, elementary first integrals. These first integrals allow one to occasionally deduce properties that an explicit solution would not necessarily reveal.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128271330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the past general purpose programs for symbol manipulation have been written for traditional Von Neumann machine architectures. The design and implementation of a simple prototype symbol manipulation system for the ICL Distributed Array Processor (DAP), is described. The system is restricted to monovariate polynomials with single precision integer coefficients. The algorithms and data structure are discussed and the design of a more general system for multivariate polynomials using arithmetic with semi-infinite precision is considered.
{"title":"On the application of Array Processors to symbol manipulation","authors":"R. Beardsworth","doi":"10.1145/800206.806382","DOIUrl":"https://doi.org/10.1145/800206.806382","url":null,"abstract":"In the past general purpose programs for symbol manipulation have been written for traditional Von Neumann machine architectures. The design and implementation of a simple prototype symbol manipulation system for the ICL Distributed Array Processor (DAP), is described. The system is restricted to monovariate polynomials with single precision integer coefficients. The algorithms and data structure are discussed and the design of a more general system for multivariate polynomials using arithmetic with semi-infinite precision is considered.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"171 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121150112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we give an extension of the Liouville theorem [RISC69, p. 169] and give a number of examples which show that integration with special functions involves some phenomena that do not occur in integration with the elementary functions alone. Our main result generalizes Liouville's theorem by allowing, in addition to the elementary functions, special functions such as the error function, Fresnel integrals and the logarithmic integral to appear in the integral of an elementary function. The basic conclusion is that these functions, if they appear, appear linearly.
本文对Liouville定理进行了推广[RISC69, p. 169]，并给出了一些例子，证明了与特殊函数积分涉及到一些单独与初等函数积分时不会出现的现象。我们的主要结果通过允许在初等函数的积分中出现误差函数、菲涅耳积分和对数积分等特殊函数来推广刘维尔定理。基本结论是，这些函数，如果出现，是线性的。
{"title":"An extension of Liouville's theorem on integration in finite terms","authors":"M. Singer, B. D. Saunders, B. Caviness","doi":"10.1137/0214069","DOIUrl":"https://doi.org/10.1137/0214069","url":null,"abstract":"In this paper we give an extension of the Liouville theorem [RISC69, p. 169] and give a number of examples which show that integration with special functions involves some phenomena that do not occur in integration with the elementary functions alone.\u0000 Our main result generalizes Liouville's theorem by allowing, in addition to the elementary functions, special functions such as the error function, Fresnel integrals and the logarithmic integral to appear in the integral of an elementary function. The basic conclusion is that these functions, if they appear, appear linearly.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"78 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115712372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper reports ongoing research at the IBM Research Center on the development of a language with extensible parameterized types and generic operators for computational algebra. The language provides an abstract data type mechanism for defining algorithms which work in as general a setting as possible. The language is based on the notions of domains and categories. Domains represent algebraic structures. Categories designate collections of domains having common operations with stated mathematical properties. Domains and categories are computed objects which may be dynamically assigned to variables, passed as arguments, and returned by functions. Although the language has been carefully tailored for the application of algebraic computation, it actually provides a very general abstract data type mechanism. Our notion of a category to group domains with common properties appears novel among programming languages (cf. image functor of RUSSELL) and leads to a very powerful notion of abstract algorithms missing from other work on data types known to the authors.
{"title":"A language for computational algebra","authors":"R. Jenks, B. Trager","doi":"10.1145/800206.806363","DOIUrl":"https://doi.org/10.1145/800206.806363","url":null,"abstract":"This paper reports ongoing research at the IBM Research Center on the development of a language with extensible parameterized types and generic operators for computational algebra. The language provides an abstract data type mechanism for defining algorithms which work in as general a setting as possible. The language is based on the notions of domains and categories. Domains represent algebraic structures. Categories designate collections of domains having common operations with stated mathematical properties. Domains and categories are computed objects which may be dynamically assigned to variables, passed as arguments, and returned by functions. Although the language has been carefully tailored for the application of algebraic computation, it actually provides a very general abstract data type mechanism. Our notion of a category to group domains with common properties appears novel among programming languages (cf. image functor of RUSSELL) and leads to a very powerful notion of abstract algorithms missing from other work on data types known to the authors.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128282178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：