In Finite Geometry, a class of objects known as BLT-sets play an important role. They are points on the Q(4,q) quadric satisfying a condition on triples. This paper is a contribution to the difficult problem of classifying these sets up to isomorphism, i.e., up to the action of the automorphism group of the quadric. We reduce the classification problem of these sets to the problem of classifying rainbow cliques in graphs. This allows us to classify BLT-sets for all orders q in the range 31 to 67.
{"title":"Rainbow cliques and the classification of small BLT-sets","authors":"Anton Betten","doi":"10.1145/2465506.2465508","DOIUrl":"https://doi.org/10.1145/2465506.2465508","url":null,"abstract":"In Finite Geometry, a class of objects known as BLT-sets play an important role. They are points on the Q(4,q) quadric satisfying a condition on triples. This paper is a contribution to the difficult problem of classifying these sets up to isomorphism, i.e., up to the action of the automorphism group of the quadric. We reduce the classification problem of these sets to the problem of classifying rainbow cliques in graphs. This allows us to classify BLT-sets for all orders q in the range 31 to 67.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121716098","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 decade there has been a surge of interest in algebraic approaches to optimization problems defined by multivariate polynomials. Fundamental mathematical challenges that arise in this area include understanding the structure of nonnegative polynomials, the interplay between efficiency and complexity of different representations of algebraic sets, and the development of effective algorithms. Remarkably, and perhaps unexpectedly, convexity provides a new viewpoint and a powerful framework for addressing these questions. This naturally brings us to the intersection of algebraic geometry, optimization, and convex geometry, with an emphasis on algorithms and computation. This emerging area has become known as convex algebraic geometry. This tutorial will focus on basic and recent developments in convex algebraic geometry, and the associated computational methods based on semidefinite programming for optimization problems involving polynomial equations and inequalities. There has been much recent progress, by combining theoretical results in real algebraic geometry with semidefinite programming to develop effective computational approaches to these problems. We will make particular emphasis on sum of squares decompositions, general duality properties, infeasibility certificates, approximation/inapproximability results, as well as survey the many exciting developments that have taken place in the last few years.
{"title":"Convex algebraic geometry and semidefinite optimization","authors":"P. Parrilo","doi":"10.1145/2465506.2466575","DOIUrl":"https://doi.org/10.1145/2465506.2466575","url":null,"abstract":"In the past decade there has been a surge of interest in algebraic approaches to optimization problems defined by multivariate polynomials. Fundamental mathematical challenges that arise in this area include understanding the structure of nonnegative polynomials, the interplay between efficiency and complexity of different representations of algebraic sets, and the development of effective algorithms. Remarkably, and perhaps unexpectedly, convexity provides a new viewpoint and a powerful framework for addressing these questions. This naturally brings us to the intersection of algebraic geometry, optimization, and convex geometry, with an emphasis on algorithms and computation. This emerging area has become known as convex algebraic geometry.\u0000 This tutorial will focus on basic and recent developments in convex algebraic geometry, and the associated computational methods based on semidefinite programming for optimization problems involving polynomial equations and inequalities. There has been much recent progress, by combining theoretical results in real algebraic geometry with semidefinite programming to develop effective computational approaches to these problems. We will make particular emphasis on sum of squares decompositions, general duality properties, infeasibility certificates, approximation/inapproximability results, as well as survey the many exciting developments that have taken place in the last few years.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"85 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126321199","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 consider the problem of efficient computations with structured polynomials. We provide complexity results for computing Fourier Transform and Truncated Fourier Transform of symmetric polynomials, and for multiplying polynomials supported on a lattice.
{"title":"Structured FFT and TFT: symmetric and lattice polynomials","authors":"J. Hoeven, R. Lebreton, É. Schost","doi":"10.1145/2465506.2465526","DOIUrl":"https://doi.org/10.1145/2465506.2465526","url":null,"abstract":"In this paper, we consider the problem of efficient computations with structured polynomials. We provide complexity results for computing Fourier Transform and Truncated Fourier Transform of symmetric polynomials, and for multiplying polynomials supported on a lattice.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125958245","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}
Proving positivity of a sequence given by a linear recurrence with polynomial coefficients (P-finite recurrence) is a non-trivial task for both humans and computers. Algorithms dealing with this task are rare or non-existent. One method that was introduced in the last decade by Gerhold and Kauers succeeds on many examples, but termination of this procedure has been proven so far only up to order three for special cases. Here we present an analysis that extends the previously known termination results on recurrences of order three, and also provides termination conditions for recurrences of higher order.
{"title":"Termination conditions for positivity proving procedures","authors":"V. Pillwein","doi":"10.1145/2465506.2465945","DOIUrl":"https://doi.org/10.1145/2465506.2465945","url":null,"abstract":"Proving positivity of a sequence given by a linear recurrence with polynomial coefficients (P-finite recurrence) is a non-trivial task for both humans and computers. Algorithms dealing with this task are rare or non-existent. One method that was introduced in the last decade by Gerhold and Kauers succeeds on many examples, but termination of this procedure has been proven so far only up to order three for special cases. Here we present an analysis that extends the previously known termination results on recurrences of order three, and also provides termination conditions for recurrences of higher order.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"139 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131781702","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}
Error-correcting decoding is generalized to multivariate sparse rational function recovery from evaluations that can be numerically inaccurate and where several evaluations can have severe errors ("outliers"). The generalization of the Berlekamp-Welch decoder to exact Cauchy interpolation of univariate rational functions from values with faults is by Kaltofen and Pernet in 2012. We give a different univariate solution based on structured linear algebra that yields a stable decoder with floating point arithmetic. Our multivariate polynomial and rational function interpolation algorithm combines Zippel's symbolic sparse polynomial interpolation technique [Ph.D. Thesis MIT 1979] with the numeric algorithm by Kaltofen, Yang, and Zhi [Proc. SNC 2007], and removes outliers ("cleans up data") through techniques from error correcting codes. Our multivariate algorithm can build a sparse model from a number of evaluations that is linear in the sparsity of the model.
{"title":"Sparse multivariate function recovery from values with noise and outlier errors","authors":"E. Kaltofen, Zhengfeng Yang","doi":"10.1145/2465506.2465524","DOIUrl":"https://doi.org/10.1145/2465506.2465524","url":null,"abstract":"Error-correcting decoding is generalized to multivariate sparse rational function recovery from evaluations that can be numerically inaccurate and where several evaluations can have severe errors (\"outliers\"). The generalization of the Berlekamp-Welch decoder to exact Cauchy interpolation of univariate rational functions from values with faults is by Kaltofen and Pernet in 2012. We give a different univariate solution based on structured linear algebra that yields a stable decoder with floating point arithmetic. Our multivariate polynomial and rational function interpolation algorithm combines Zippel's symbolic sparse polynomial interpolation technique [Ph.D. Thesis MIT 1979] with the numeric algorithm by Kaltofen, Yang, and Zhi [Proc. SNC 2007], and removes outliers (\"cleans up data\") through techniques from error correcting codes. Our multivariate algorithm can build a sparse model from a number of evaluations that is linear in the sparsity of the model.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129441141","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 tutorial surveys state-of-the-art algorithms and computational methods for computing exact solutions to linear and mixed-integer programming problems.
本教程介绍了用于计算线性和混合整数规划问题精确解的最新算法和计算方法。
{"title":"Exact linear and integer programming: tutorial abstract","authors":"Daniel E. Steffy","doi":"10.1145/2465506.2465931","DOIUrl":"https://doi.org/10.1145/2465506.2465931","url":null,"abstract":"This tutorial surveys state-of-the-art algorithms and computational methods for computing exact solutions to linear and mixed-integer programming problems.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129734255","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 propose two algorithms for verifying the existence of real solutions of positive-dimensional polynomial systems. The first one is based on the critical point method and the homotopy continuation method. It targets for verifying the existence of real roots on each connected component of an algebraic variety V ∩ Rn defined by polynomial equations. The second one is based on the low-rank moment matrix completion method and aims for verifying the existence of at least one real roots on V ∩ Rn. Combined both algorithms with the verification algorithms for zero-dimensional polynomial systems, we are able to find verified real solutions of positive-dimensional polynomial systems very efficiently for a large set of examples.
{"title":"Verified error bounds for real solutions of positive-dimensional polynomial systems","authors":"Zhengfeng Yang, L. Zhi, Yijun Zhu","doi":"10.1145/2465506.2465951","DOIUrl":"https://doi.org/10.1145/2465506.2465951","url":null,"abstract":"In this paper, we propose two algorithms for verifying the existence of real solutions of positive-dimensional polynomial systems. The first one is based on the critical point method and the homotopy continuation method. It targets for verifying the existence of real roots on each connected component of an algebraic variety V ∩ Rn defined by polynomial equations. The second one is based on the low-rank moment matrix completion method and aims for verifying the existence of at least one real roots on V ∩ Rn. Combined both algorithms with the verification algorithms for zero-dimensional polynomial systems, we are able to find verified real solutions of positive-dimensional polynomial systems very efficiently for a large set of examples.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"152 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133686306","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}
Given a matrix of univariate polynomials over a field K, its columns generate a K[x]-module. We call any basis of this module a column basis of the given matrix. Matrix gcds and matrix normal forms are examples of such module bases. In this paper we present a deterministic algorithm for the computation of a column basis of an m x n input matrix with m ≤ n. If s is the average column degree of the input matrix, this algorithm computes a column basis with a cost of Õ(nmω-1s) field operations in K. Here the soft-O notation is Big-O with log factors removed while ω is the exponent of matrix multiplication. Note that the average column degree s is bounded by the commonly used matrix degree that is also the maximum column degree of the input matrix.
{"title":"Computing column bases of polynomial matrices","authors":"Wei Zhou, G. Labahn","doi":"10.1145/2465506.2465947","DOIUrl":"https://doi.org/10.1145/2465506.2465947","url":null,"abstract":"Given a matrix of univariate polynomials over a field <i>K</i>, its columns generate a <i>K</i>[<i>x</i>]-module. We call any basis of this module a column basis of the given matrix. Matrix gcds and matrix normal forms are examples of such module bases. In this paper we present a deterministic algorithm for the computation of a column basis of an <i>m</i> x <i>n</i> input matrix with <i>m</i> ≤ <i>n</i>. If <i>s</i> is the average column degree of the input matrix, this algorithm computes a column basis with a cost of Õ(<i>nm</i><sup>ω-1</sup>s) field operations in <i>K</i>. Here the soft-<i>O</i> notation is Big-<i>O</i> with log factors removed while ω is the exponent of matrix multiplication. Note that the average column degree <i>s</i> is bounded by the commonly used matrix degree that is also the maximum column degree of the input matrix.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129268491","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}
Laureano Lambán, F. Martín-Mateos, J. Rubio, J. Ruiz-Reina
Certified symbolic manipulation is an emerging new field where programs are accompanied by certificates that, suitably interpreted, ensure the correctness of the algorithms. In this paper, we focus on algebraic algorithms implemented in the proof assistant ACL2, which allows us to verify correctness in the same programming environment. The case study is that of bivariate simplicial polynomials, a data structure used to help the proof of properties in Simplicial Topology. Simplicial polynomials can be computationally interpreted in two ways. As symbolic expressions, they can be handled algorithmically, increasing the automation in ACL2 proofs. As representations of functional operators, they help proving properties of categorical morphisms. As an application of this second view, we present the definition in ACL2 of some morphisms involved in the Eilenberg-Zilber reduction, a central part of the Kenzo computer algebra system. We have proved the ACL2 implementations are correct and tested that they get the same results as Kenzo does.
{"title":"Certified symbolic manipulation: bivariate simplicial polynomials","authors":"Laureano Lambán, F. Martín-Mateos, J. Rubio, J. Ruiz-Reina","doi":"10.1145/2465506.2465515","DOIUrl":"https://doi.org/10.1145/2465506.2465515","url":null,"abstract":"Certified symbolic manipulation is an emerging new field where programs are accompanied by certificates that, suitably interpreted, ensure the correctness of the algorithms. In this paper, we focus on algebraic algorithms implemented in the proof assistant ACL2, which allows us to verify correctness in the same programming environment. The case study is that of bivariate simplicial polynomials, a data structure used to help the proof of properties in Simplicial Topology. Simplicial polynomials can be computationally interpreted in two ways. As symbolic expressions, they can be handled algorithmically, increasing the automation in ACL2 proofs. As representations of functional operators, they help proving properties of categorical morphisms. As an application of this second view, we present the definition in ACL2 of some morphisms involved in the Eilenberg-Zilber reduction, a central part of the Kenzo computer algebra system. We have proved the ACL2 implementations are correct and tested that they get the same results as Kenzo does.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130460657","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 HJLS and PSLQ algorithms are the de facto standards for discovering non-trivial integer relations between a given tuple of real numbers. In this work, we provide a new interpretation of these algorithms, in a more general and powerful algebraic setup: we view them as special cases of algorithms that compute the intersection between a lattice and a vector subspace. Further, we extract from them the first algorithm for manipulating finitely generated additive subgroups of a euclidean space, including projections of lattices and finite sums of lattices. We adapt the analyses of HJLS and PSLQ to derive correctness and convergence guarantees.
{"title":"A new view on HJLS and PSLQ: sums and projections of lattices","authors":"Jingwei Chen, D. Stehlé, G. Villard","doi":"10.1145/2465506.2465936","DOIUrl":"https://doi.org/10.1145/2465506.2465936","url":null,"abstract":"The HJLS and PSLQ algorithms are the de facto standards for discovering non-trivial integer relations between a given tuple of real numbers. In this work, we provide a new interpretation of these algorithms, in a more general and powerful algebraic setup: we view them as special cases of algorithms that compute the intersection between a lattice and a vector subspace. Further, we extract from them the first algorithm for manipulating finitely generated additive subgroups of a euclidean space, including projections of lattices and finite sums of lattices. We adapt the analyses of HJLS and PSLQ to derive correctness and convergence guarantees.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"25 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113984685","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：