Rui-Juan Jing, M. M. Maza, Yan-Feng Xie, Chun-Ming Yuan
{"title":"Efficient detection of redundancies in systems of linear inequalities✱","authors":"Rui-Juan Jing, M. M. Maza, Yan-Feng Xie, Chun-Ming Yuan","doi":"10.1145/3666000.3669708","DOIUrl":"https://doi.org/10.1145/3666000.3669708","url":null,"abstract":"","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"6 11","pages":"351-360"},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141640832","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 deal with the problem of simultaneous reconstruction of a vector of rational numbers, given modular reductions containing errors (SRNRwE). Our methods apply as well to the simultaneous reconstruction of rational functions given evaluations containing errors (SRFRwE), improving known results [7, 9]. In the latter case, one can take advantage of techniques from coding theory [4, 10] and provide an algorithm that extends classical Reed-Solomon decoding. In recent works [7, 9], interleaved Reed-Solomon codes [3, 19] are used to correct beyond the unique decoding capability in the case of random errors at the price of positive but small failure probability. Our first contribution is to extend these works to the simultaneous reconstruction with errors of rational numbers instead of functions. Thus considering rational number codes [16], we provide an algorithm decoding beyond the unique decoding capability and, as a central result of this paper, we analyze in detail its failure probability. Our analysis generalizes for the first time the best known analysis for interleaved Reed-Solomon codes [19] to SRFRwE, improving on the existing bound [8], to interleaved Chinese remainder codes, also improving the known bound [1], and finally for the first time to SRNRwE.
{"title":"Decoding Simultaneous Rational Evaluation Codes","authors":"Matteo Abbondati, Eleonora Guerrini, R. Lebreton","doi":"10.1145/3666000.3669686","DOIUrl":"https://doi.org/10.1145/3666000.3669686","url":null,"abstract":"In this paper, we deal with the problem of simultaneous reconstruction of a vector of rational numbers, given modular reductions containing errors (SRNRwE). Our methods apply as well to the simultaneous reconstruction of rational functions given evaluations containing errors (SRFRwE), improving known results [7, 9]. In the latter case, one can take advantage of techniques from coding theory [4, 10] and provide an algorithm that extends classical Reed-Solomon decoding. In recent works [7, 9], interleaved Reed-Solomon codes [3, 19] are used to correct beyond the unique decoding capability in the case of random errors at the price of positive but small failure probability. Our first contribution is to extend these works to the simultaneous reconstruction with errors of rational numbers instead of functions. Thus considering rational number codes [16], we provide an algorithm decoding beyond the unique decoding capability and, as a central result of this paper, we analyze in detail its failure probability. Our analysis generalizes for the first time the best known analysis for interleaved Reed-Solomon codes [19] to SRFRwE, improving on the existing bound [8], to interleaved Chinese remainder codes, also improving the known bound [1], and finally for the first time to SRNRwE.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"1 5","pages":"153-161"},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141641776","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}
{"title":"Enumerating polynomial colored permutation classes","authors":"Saúl A. Blanco, Daniel E. Skora","doi":"10.1145/3666000.3669700","DOIUrl":"https://doi.org/10.1145/3666000.3669700","url":null,"abstract":"","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"1 3","pages":"283-291"},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141642457","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}
{"title":"Automated Reasoning For The Existence Of Darboux Polynomials","authors":"Khalil Ghorbal, Maxime Bridoux","doi":"10.1145/3666000.3669705","DOIUrl":"https://doi.org/10.1145/3666000.3669705","url":null,"abstract":"","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"8 17","pages":"324-333"},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141640434","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}
Finding closed form solutions of differential equations has a long history in computer algebra. For example, the Risch algorithm (1969) decides if the equation y' = f can be solved in terms of elementary functions. These are functions that can be written in terms of exp and log, where "in terms of" allows for field operations, composition, and algebraic extensions. More generally, functions are in closed form if they are written in terms of commonly used functions. This includes not only exp and log, but other common functions as well, such as Bessel functions or the Gauss hypergeometric function. Given a differential equation L, to find solutions written in terms of such functions, one seeks a sequence of transformations that sends the Bessel equation, or the Gauss hypergeometric equation, to L. Although random equations are unlikely to have closed form solutions, they are remarkably common in applications. For example, if y = ∑n=0∞ an xn has a positive radius of convergence, integer coefficients an, and satisfies a second order homogeneous linear differential equation L with polynomial coefficients, then L is conjectured to be solvable in closed form. Such equations are common, not only in combinatorics, but in physics as well. The talk will describe recent progress in finding closed form solutions of differential and difference equations, as well as open questions.
{"title":"Closed Form Solutions for Linear Differential and Difference Equations","authors":"M. V. Hoeij","doi":"10.1145/3087604.3087660","DOIUrl":"https://doi.org/10.1145/3087604.3087660","url":null,"abstract":"Finding closed form solutions of differential equations has a long history in computer algebra. For example, the Risch algorithm (1969) decides if the equation y' = f can be solved in terms of elementary functions. These are functions that can be written in terms of exp and log, where \"in terms of\" allows for field operations, composition, and algebraic extensions. More generally, functions are in closed form if they are written in terms of commonly used functions. This includes not only exp and log, but other common functions as well, such as Bessel functions or the Gauss hypergeometric function. Given a differential equation L, to find solutions written in terms of such functions, one seeks a sequence of transformations that sends the Bessel equation, or the Gauss hypergeometric equation, to L. Although random equations are unlikely to have closed form solutions, they are remarkably common in applications. For example, if y = ∑n=0∞ an xn has a positive radius of convergence, integer coefficients an, and satisfies a second order homogeneous linear differential equation L with polynomial coefficients, then L is conjectured to be solvable in closed form. Such equations are common, not only in combinatorics, but in physics as well. The talk will describe recent progress in finding closed form solutions of differential and difference equations, as well as open questions.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125398348","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}
Pub Date : 2015-09-08DOI: 10.1007/3-540-51084-2_39
D. Cantone, V. Cutello, A. Ferro
{"title":"Decision Procedures for Elementary Sublanguages of Set Theory. XIV. Three Languages Involving Rank Related Constructs","authors":"D. Cantone, V. Cutello, A. Ferro","doi":"10.1007/3-540-51084-2_39","DOIUrl":"https://doi.org/10.1007/3-540-51084-2_39","url":null,"abstract":"","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123480705","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}
A projection operator of a system of parametric polynomials is a polynomial in the coefficients of the system that vanishes if the system has a common root. The projection operator is a multiple of the resultant of the system, and the factors of the projection operator that are not contained in the resultant are called extraneous factors. The main contribution of this work is to provide a randomized algorithm to check whether a factor is extraneous, which is an important task in applications. A lower bound for the success probability is determined which can be set arbitrarily close to one. This algorithm uses certain matrices rather than Gröbner bases and seems to be the first algorithm of this kind for this task.
{"title":"Randomized detection of extraneous factors","authors":"Manfred Minimair","doi":"10.1145/2608628.2608631","DOIUrl":"https://doi.org/10.1145/2608628.2608631","url":null,"abstract":"A projection operator of a system of parametric polynomials is a polynomial in the coefficients of the system that vanishes if the system has a common root. The projection operator is a multiple of the resultant of the system, and the factors of the projection operator that are not contained in the resultant are called extraneous factors. The main contribution of this work is to provide a randomized algorithm to check whether a factor is extraneous, which is an important task in applications. A lower bound for the success probability is determined which can be set arbitrarily close to one. This algorithm uses certain matrices rather than Gröbner bases and seems to be the first algorithm of this kind for this task.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"237 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126902182","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 article describes a Mathematica package that improves simplification of general non-numeric expressions containing any mixture of Gaussian rational numbers, symbolic constants, machine and arbitrary-precision floating-point numbers, together with intervals having any mixture of such endpoints. Such generalized numbers are not automatically all converted to floats or to intervals. Expressions can be multivariate and non-polynomial. Techniques include: • Recognition and unification of approximately similar and approximately proportional factors and terms. • The option of infinity-norm normalization that is more robust than monic normalization. • The option of a unit-normal quasi-primitive normalization that uses large rational approximate common divisors of mixed number types and intervals to nicely normalize sums. • A polynomial division algorithm tolerant of terms with coefficients that are float zeros or intervals containing 0. • The ability to round or underflow negligible terms while satisfying the inclusion property of interval arithmetic. The package and a more detailed version of this article will be posted at arXiv.org.
{"title":"Fuzzy simplification of non-numeric expressions containing some intervals and/or floating point numbers","authors":"D. R. Stoutemyer","doi":"10.1145/2608628.2627489","DOIUrl":"https://doi.org/10.1145/2608628.2627489","url":null,"abstract":"This article describes a Mathematica package that improves simplification of general non-numeric expressions containing any mixture of Gaussian rational numbers, symbolic constants, machine and arbitrary-precision floating-point numbers, together with intervals having any mixture of such endpoints. Such generalized numbers are not automatically all converted to floats or to intervals. Expressions can be multivariate and non-polynomial. Techniques include:\u0000 • Recognition and unification of approximately similar and approximately proportional factors and terms.\u0000 • The option of infinity-norm normalization that is more robust than monic normalization.\u0000 • The option of a unit-normal quasi-primitive normalization that uses large rational approximate common divisors of mixed number types and intervals to nicely normalize sums.\u0000 • A polynomial division algorithm tolerant of terms with coefficients that are float zeros or intervals containing 0.\u0000 • The ability to round or underflow negligible terms while satisfying the inclusion property of interval arithmetic.\u0000 The package and a more detailed version of this article will be posted at arXiv.org.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114757371","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}
Semi-mixed algebraic systems are those where the equations can be partitioned into subsets with common Newton polytopes. We are interested in counting roots of semi-mixed multihomogeneous systems, where both variables and equations can be partitioned into blocks, and each block of equations has a given degree in each block of variables. The motivating example is counting the number of totally mixed Nash equilibria in games of several players. Firstly, this paper relates and unifies the BKK and multivariate Bézout bounds for semi-mixed systems, through mixed volumes and matrix permanents. Permanent expressions for BKK bounds hold for all multihomogeneous systems, without any requirement of semi-mixed structure, as well as systems whose Newton polytopes are products of polytopes in complementary subspaces. Secondly, by means of a novel asymptotic analysis, the complexity of a combinatorial geometric algorithm for semi-mixed volumes (i.e., mixed volumes of semi-mixed systems) is explored and juxtaposed to the complexities of computing permanents, or using generating functions (via MacMahon's Master theorem), or orthogonal polynomials.
{"title":"Root counts of semi-mixed systems, and an application to counting nash equilibria","authors":"I. Emiris, R. Vidunas","doi":"10.1145/2608628.2608679","DOIUrl":"https://doi.org/10.1145/2608628.2608679","url":null,"abstract":"Semi-mixed algebraic systems are those where the equations can be partitioned into subsets with common Newton polytopes. We are interested in counting roots of semi-mixed multihomogeneous systems, where both variables and equations can be partitioned into blocks, and each block of equations has a given degree in each block of variables. The motivating example is counting the number of totally mixed Nash equilibria in games of several players. Firstly, this paper relates and unifies the BKK and multivariate Bézout bounds for semi-mixed systems, through mixed volumes and matrix permanents. Permanent expressions for BKK bounds hold for all multihomogeneous systems, without any requirement of semi-mixed structure, as well as systems whose Newton polytopes are products of polytopes in complementary subspaces. Secondly, by means of a novel asymptotic analysis, the complexity of a combinatorial geometric algorithm for semi-mixed volumes (i.e., mixed volumes of semi-mixed systems) is explored and juxtaposed to the complexities of computing permanents, or using generating functions (via MacMahon's Master theorem), or orthogonal polynomials.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134003578","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 [Kaltofen and Yang, Proc. ISSAC 2013] we have generalized algebraic error-correcting decoding to multivariate sparse rational function interpolation from evaluations that can be numerically inaccurate and where several evaluations can have severe errors ("outliers"). Here we present a different algorithm that can interpolate a sparse multivariate rational function from evaluations where the error rate is 1/q for any q > 2, which our ISSAC 2013 algorithm could not handle. When implemented as a numerical algorithm we can, for instance, reconstruct a fraction of trinomials of degree 15 in 50 variables with non-outlier evaluations of relative noise as large as 10-7 and where as much as 1/4 of the 14717 evaluations are outliers with relative error as small as 0.01 (large outliers are easily located by our method). For the algorithm with exact arithmetic and exact values at non-erroneous points, we provide a proof that for random evaluations one can avoid quadratic oversampling. Our argument already applies to our original 2007 sparse rational function interpolation algorithm [Kaltofen, Yang and Zhi, Proc. SNC 2007], where we have experimentally observed that for T unknown non-zero coefficients in a sparse candidate ansatz one only needs T +O(1) evaluations rather than the proven O(T2) (cf. Candès and Tao sparse sensing). Here we finally can give the probabilistic analysis for this fact.
在[Kaltofen和Yang, Proc. ISSAC 2013]中，我们将代数错误纠正解码推广到多元稀疏有理函数插值，这些插值来自可能在数值上不准确的评估，其中几个评估可能有严重的错误(“异常值”)。在这里，我们提出了一种不同的算法，可以从错误率为1/q的评估中插值一个稀疏的多元理性函数，对于任何q > 2，我们的ISSAC 2013算法无法处理。例如，当作为数值算法实现时，我们可以重建50个变量中15度三项式的一小部分，其相对噪声的非离群值评估高达10-7，其中14717个评估中多达1/4是相对误差小至0.01的离群值(通过我们的方法很容易找到大的离群值)。对于具有精确算术和精确值的非错误点算法，我们证明了对于随机求值可以避免二次过采样。我们的论点已经适用于我们最初的2007年稀疏有理函数插值算法[Kaltofen, Yang和Zhi, Proc. SNC 2007]，我们在实验中观察到，对于稀疏候选ansatz中的T个未知非零系数，只需要T +O(1)次评估，而不是证明的O(T2)次评估(参见cand和Tao稀疏感知)。这里我们终于可以给出这个事实的概率分析。
{"title":"Sparse multivariate function recovery with a high error rate in the evaluations","authors":"E. Kaltofen, Zhengfeng Yang","doi":"10.1145/2608628.2608637","DOIUrl":"https://doi.org/10.1145/2608628.2608637","url":null,"abstract":"In [Kaltofen and Yang, Proc. ISSAC 2013] we have generalized algebraic error-correcting decoding to multivariate sparse rational function interpolation from evaluations that can be numerically inaccurate and where several evaluations can have severe errors (\"outliers\"). Here we present a different algorithm that can interpolate a sparse multivariate rational function from evaluations where the error rate is 1/q for any q > 2, which our ISSAC 2013 algorithm could not handle. When implemented as a numerical algorithm we can, for instance, reconstruct a fraction of trinomials of degree 15 in 50 variables with non-outlier evaluations of relative noise as large as 10-7 and where as much as 1/4 of the 14717 evaluations are outliers with relative error as small as 0.01 (large outliers are easily located by our method).\u0000 For the algorithm with exact arithmetic and exact values at non-erroneous points, we provide a proof that for random evaluations one can avoid quadratic oversampling. Our argument already applies to our original 2007 sparse rational function interpolation algorithm [Kaltofen, Yang and Zhi, Proc. SNC 2007], where we have experimentally observed that for T unknown non-zero coefficients in a sparse candidate ansatz one only needs T +O(1) evaluations rather than the proven O(T2) (cf. Candès and Tao sparse sensing). Here we finally can give the probabilistic analysis for this fact.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129580231","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}
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