The Apagodu-Zeilberger algorithm can be used for computing annihilating operators for definite sums over hypergeometric terms, or for definite integrals over hyperexponential functions. In this paper, we propose a generalization of this algorithm which is applicable to arbitrary Δ-finite functions. In analogy to the hypergeometric case, we introduce the notion of proper Δ-finite functions. We show that the algorithm always succeeds for these functions, and we give a tight a priori bound for the order of the output operator.
{"title":"A generalized Apagodu-Zeilberger algorithm","authors":"Shaoshi Chen, Manuel Kauers, C. Koutschan","doi":"10.1145/2608628.2608641","DOIUrl":"https://doi.org/10.1145/2608628.2608641","url":null,"abstract":"The Apagodu-Zeilberger algorithm can be used for computing annihilating operators for definite sums over hypergeometric terms, or for definite integrals over hyperexponential functions. In this paper, we propose a generalization of this algorithm which is applicable to arbitrary Δ-finite functions. In analogy to the hypergeometric case, we introduce the notion of proper Δ-finite functions. We show that the algorithm always succeeds for these functions, and we give a tight a priori bound for the order of the output operator.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128612117","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 will give an overview of algebraic complexity theory focused on bilinear complexity, and describe several powerful techniques to analyze the complexity of computational problems from linear algebra, in particular matrix multiplication. The presentation of these techniques will follow the history of progress on constructing asymptotically fast algorithms for matrix multiplication, and include its most recent developments.
{"title":"Algebraic complexity theory and matrix multiplication","authors":"F. Gall","doi":"10.1145/2608628.2627493","DOIUrl":"https://doi.org/10.1145/2608628.2627493","url":null,"abstract":"This tutorial will give an overview of algebraic complexity theory focused on bilinear complexity, and describe several powerful techniques to analyze the complexity of computational problems from linear algebra, in particular matrix multiplication. The presentation of these techniques will follow the history of progress on constructing asymptotically fast algorithms for matrix multiplication, and include its most recent developments.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"26 6‐7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120851828","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}
Óscar García-Morchón, R. Rietman, L. Tolhuizen, Domingo Gómez-Pérez, J. Gutierrez
We consider a two polynomials analogue of the polynomial interpolation problem. Namely, we consider the Mixing Modular Operations (MMO) problem of recovering two polynomials f ∈ Zp[x] and g ∈ Zq[x] of known degree, where p and q are two (un)known positive integers, from the values of f(t) mod p+g(t) mod q at polynomially many points t ∈ Z. We show that if p and q are known, the MMO problem can be reduced to computing a close vector in a lattice with respect to the infinity norm. Using the Gaussian heuristic we also implemented in the SAGE system a polynomial-time algorithm. If p and q are kept secret, we do not know how to solve this problem. This problem is motivated by several potential cryptographic applications.
我们考虑一个多项式插值问题的双多项式模拟。也就是说，我们考虑从多项式多个点t∈z处的f(t) mod p+g(t) mod q的值中恢复两个已知阶多项式f∈Zp[x]和g∈Zq[x]的混合模操作(MMO)问题。我们证明，如果p和q是已知的，MMO问题可以简化为计算晶格中关于无穷范数的接近向量。我们还利用高斯启发式算法在SAGE系统中实现了一个多项式时间算法。如果p和q是保密的，我们不知道如何解决这个问题。这个问题是由几个潜在的加密应用程序引起的。
{"title":"The MMO problem","authors":"Óscar García-Morchón, R. Rietman, L. Tolhuizen, Domingo Gómez-Pérez, J. Gutierrez","doi":"10.1145/2608628.2608643","DOIUrl":"https://doi.org/10.1145/2608628.2608643","url":null,"abstract":"We consider a two polynomials analogue of the polynomial interpolation problem. Namely, we consider the Mixing Modular Operations (MMO) problem of recovering two polynomials <i>f</i> ∈ Z<sub><i>p</i></sub>[<i>x</i>] and <i>g</i> ∈ Z<sub><i>q</i></sub>[<i>x</i>] of known degree, where <i>p</i> and <i>q</i> are two (un)known positive integers, from the values of <i>f</i>(<i>t</i>) mod <i>p</i>+<i>g</i>(<i>t</i>) mod <i>q</i> at polynomially many points <i>t</i> ∈ Z. We show that if <i>p</i> and <i>q</i> are known, the MMO problem can be reduced to computing a close vector in a lattice with respect to the infinity norm. Using the Gaussian heuristic we also implemented in the SAGE system a polynomial-time algorithm. If <i>p</i> and <i>q</i> are kept secret, we do not know how to solve this problem. This problem is motivated by several potential cryptographic applications.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"101 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115639416","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}
We present new techniques for reducing a multivariate sparse polynomial to a univariate polynomial. The reduction works similarly to the classical and widely-used Kronecker substitution, except that we choose the degrees randomly based on the number of nonzero terms in the multivariate polynomial. The resulting univariate polynomial often has a significantly lower degree than the Kronecker substitution polynomial, at the expense of a small number of term collisions. As an application, we give a new algorithm for multivariate interpolation which uses these new techniques along with any existing univariate interpolation algorithm.
{"title":"Multivariate sparse interpolation using randomized Kronecker substitutions","authors":"A. Arnold, Daniel S. Roche","doi":"10.1145/2608628.2608674","DOIUrl":"https://doi.org/10.1145/2608628.2608674","url":null,"abstract":"We present new techniques for reducing a multivariate sparse polynomial to a univariate polynomial. The reduction works similarly to the classical and widely-used Kronecker substitution, except that we choose the degrees randomly based on the number of nonzero terms in the multivariate polynomial. The resulting univariate polynomial often has a significantly lower degree than the Kronecker substitution polynomial, at the expense of a small number of term collisions. As an application, we give a new algorithm for multivariate interpolation which uses these new techniques along with any existing univariate interpolation algorithm.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115264004","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 an algorithm for computing a fundamental matrix of formal solutions of completely integrable Pfaffian systems with normal crossings in two variables. First, we associate to the Pfaffian system a singular linear system of ordinary differential equations from which its formal invariants can be efficiently derived. After that, we give a generalization of the Moser-based rank reduction algorithm of [5]. These two items allow us to construct formal solutions by following the recursive algorithm given in [4] for singular linear systems of ordinary differential equations. Our algorithm builds upon the package ISOLDE [9] and is implemented in the computer algebra system Maple.
{"title":"Formal solutions of a class of Pfaffian systems in two variables","authors":"Suzy S. Maddah, M. Barkatou, H. Abbas","doi":"10.1145/2608628.2608656","DOIUrl":"https://doi.org/10.1145/2608628.2608656","url":null,"abstract":"In this paper, we present an algorithm for computing a fundamental matrix of formal solutions of completely integrable Pfaffian systems with normal crossings in two variables. First, we associate to the Pfaffian system a singular linear system of ordinary differential equations from which its formal invariants can be efficiently derived. After that, we give a generalization of the Moser-based rank reduction algorithm of [5]. These two items allow us to construct formal solutions by following the recursive algorithm given in [4] for singular linear systems of ordinary differential equations. Our algorithm builds upon the package ISOLDE [9] and is implemented in the computer algebra system Maple.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"111 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131736993","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 article, we treat the turning points of singularly-perturbed linear differential systems and reduce their parameter singularity's rank to its minimal integer value. Our approach is Moser-based, i.e. it is based on the reduction criterion introduced for singular linear differential systems by Moser [21]. Such algorithms have proved their utility in the symbolic resolution of the systems of linear functional equations [5, 6, 8], giving rise to the package ISOLDE [7], as well as in the perturbed algebraic eigenvalue problem [13]. In particular, we generalize the Moser-based algorithm described in [4]. Our algorithm, implemented in the computer algebra system Maple, paves the way for efficient symbolic resolution of singularly-perturbed linear differential systems as well as further applications of Moser-based reduction over bivariate (differential) fields [1].
{"title":"On the reduction of singularly-perturbed linear differential systems","authors":"Suzy S. Maddah, M. Barkatou, H. Abbas","doi":"10.1145/2608628.2608655","DOIUrl":"https://doi.org/10.1145/2608628.2608655","url":null,"abstract":"In this article, we treat the turning points of singularly-perturbed linear differential systems and reduce their parameter singularity's rank to its minimal integer value. Our approach is Moser-based, i.e. it is based on the reduction criterion introduced for singular linear differential systems by Moser [21]. Such algorithms have proved their utility in the symbolic resolution of the systems of linear functional equations [5, 6, 8], giving rise to the package ISOLDE [7], as well as in the perturbed algebraic eigenvalue problem [13]. In particular, we generalize the Moser-based algorithm described in [4]. Our algorithm, implemented in the computer algebra system Maple, paves the way for efficient symbolic resolution of singularly-perturbed linear differential systems as well as further applications of Moser-based reduction over bivariate (differential) fields [1].","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125072934","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}
We develop algorithms to compute the differential Galois group G associated to a parameterized second-order homogeneous linear differential equation of the form [EQUATION] where the coefficients r1, r0 ∈ F(x) are rational functions in x with coefficients in a partial differential field F of characteristic zero. This work relies on earlier procedures developed by Dreyfus and by the present author to compute G when r1 = 0. By reinterpreting a classical change-of-variables procedure in Galois-theoretic terms, we complete these algorithms to compute G with no restrictions on r1.
{"title":"Computing the differential Galois group of a parameterized second-order linear differential equation","authors":"Carlos E. Arreche","doi":"10.1145/2608628.2608680","DOIUrl":"https://doi.org/10.1145/2608628.2608680","url":null,"abstract":"We develop algorithms to compute the differential Galois group <i>G</i> associated to a parameterized second-order homogeneous linear differential equation of the form\u0000 [EQUATION]\u0000 where the coefficients <i>r</i><sub>1</sub>, <i>r</i><sub>0</sub> ∈ <i>F</i>(<i>x</i>) are rational functions in <i>x</i> with coefficients in a partial differential field <i>F</i> of characteristic zero. This work relies on earlier procedures developed by Dreyfus and by the present author to compute <i>G</i> when <i>r</i><sub>1</sub> = 0. By reinterpreting a classical change-of-variables procedure in Galois-theoretic terms, we complete these algorithms to compute <i>G</i> with no restrictions on <i>r</i><sub>1</sub>.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128567006","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}
We present a new Monte Carlo algorithm for the interpolation of a straight-line program as a sparse polynomial f over an arbitrary finite field of size q. We assume a priori bounds D and T are given on the degree and number of terms of f. The approach presented in this paper is a hybrid of the diversified and recursive interpolation algorithms, the two previous fastest known probabilistic methods for this problem. By making effective use of the information contained in the coefficients themselves, this new algorithm improves on the bit complexity of previous methods by a "soft-Oh" factor of T, log D, or log q.
{"title":"Sparse interpolation over finite fields via low-order roots of unity","authors":"A. Arnold, M. Giesbrecht, Daniel S. Roche","doi":"10.1145/2608628.2608671","DOIUrl":"https://doi.org/10.1145/2608628.2608671","url":null,"abstract":"We present a new Monte Carlo algorithm for the interpolation of a straight-line program as a sparse polynomial f over an arbitrary finite field of size q. We assume a priori bounds D and T are given on the degree and number of terms of f. The approach presented in this paper is a hybrid of the diversified and recursive interpolation algorithms, the two previous fastest known probabilistic methods for this problem. By making effective use of the information contained in the coefficients themselves, this new algorithm improves on the bit complexity of previous methods by a \"soft-Oh\" factor of T, log D, or log q.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"16 5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116791443","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}
Certificates to a linear algebra computation are additional data structures for each output, which can be used by a---possibly randomized---verification algorithm that proves the correctness of each output. The certificates are essentially optimal if the time (and space) complexity of verification is essentially linear in the input size N, meaning N times a factor No(1), i.e., a factor Nη(N) with limN → ∞ η(N) = 0. We give algorithms that compute essentially optimal certificates for the positive semidefiniteness, Frobenius form, characteristic and minimal polynomial of an n × n dense integer matrix A. Our certificates can be verified in Monte-Carlo bit complexity (n2 log ||A||)1+o(1), where log ||A|| is the bit size of the integer entries, solving an open problem in [Kaltofen, Nehring, Saunders, Proc. ISSAC 2011] subject to computational hardness assumptions. Second, we give algorithms that compute certificates for the rank of sparse or structured n × n matrices over an abstract field, whose Monte Carlo verification complexity is 2 matrix-times-vector products + n1+o(1) arithmetic operations in the field. For example, if the n × n input matrix is sparse with n1+o(1) non-zero entries, our rank certificate can be verified in n1+o(1) field operations. This extends also to integer matrices with only an extra log ||A||1+o(1) factor. All our certificates are based on interactive verification protocols with the interaction removed by a Fiat-Shamir identification heuristic. The validity of our verification procedure is subject to standard computational hardness assumptions from cryptography.
{"title":"Essentially optimal interactive certificates in linear algebra","authors":"J. Dumas, E. Kaltofen","doi":"10.1145/2608628.2608644","DOIUrl":"https://doi.org/10.1145/2608628.2608644","url":null,"abstract":"Certificates to a linear algebra computation are additional data structures for each output, which can be used by a---possibly randomized---verification algorithm that proves the correctness of each output. The certificates are essentially optimal if the time (and space) complexity of verification is essentially linear in the input size <i>N</i>, meaning <i>N</i> times a factor <i>N</i><sup><i>o</i>(1)</sup>, i.e., a factor <i>N</i><sup><i>η</i>(<i>N</i>)</sup> with lim<sub><i>N</i> → ∞</sub> η(<i>N</i>) = 0.\u0000 We give algorithms that compute essentially optimal certificates for the positive semidefiniteness, Frobenius form, characteristic and minimal polynomial of an <i>n × n</i> dense integer matrix <i>A</i>. Our certificates can be verified in Monte-Carlo bit complexity (<i>n</i><sup>2</sup> log ||<i>A</i>||)<sup>1+o(1)</sup>, where log ||A|| is the bit size of the integer entries, solving an open problem in [Kaltofen, Nehring, Saunders, Proc. ISSAC 2011] subject to computational hardness assumptions.\u0000 Second, we give algorithms that compute certificates for the rank of sparse or structured <i>n × n</i> matrices over an abstract field, whose Monte Carlo verification complexity is 2 matrix-times-vector products + <i>n</i><sup>1+o(1)</sup> arithmetic operations in the field. For example, if the <i>n × n</i> input matrix is sparse with <i>n</i><sup>1+o(1)</sup> non-zero entries, our rank certificate can be verified in <i>n</i><sup>1+o(1)</sup> field operations. This extends also to integer matrices with only an extra log ||<i>A</i>||<sup>1+o(1)</sup> factor.\u0000 All our certificates are based on interactive verification protocols with the interaction removed by a Fiat-Shamir identification heuristic. The validity of our verification procedure is subject to standard computational hardness assumptions from cryptography.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"122 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131162672","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 has been shown recently that monomial maps in a large class respecting the action of the infinite symmetric group have, up to symmetry, finitely generated kernels. We study the simplest nontrivial family in this class: the maps given by a single monomial. Considering the corresponding lattice map, we explicitly construct an equivariant lattice generating set, whose width (the number of variables necessary to write it down) depends linearly on the width of the map. This result is sharp and improves dramatically the previously known upper bound as it does not depend on the degree of the image monomial. In the case of of width two, we construct an explicit finite set of binomials generating the toric ideal up to symmetry. Both width and degree of this generating set are sharply bounded by linear functions in the exponents of the monomial.
{"title":"Equivariant lattice generators and Markov bases","authors":"Thomas Kahle, Robert Krone, A. Leykin","doi":"10.1145/2608628.2608646","DOIUrl":"https://doi.org/10.1145/2608628.2608646","url":null,"abstract":"It has been shown recently that monomial maps in a large class respecting the action of the infinite symmetric group have, up to symmetry, finitely generated kernels. We study the simplest nontrivial family in this class: the maps given by a single monomial. Considering the corresponding lattice map, we explicitly construct an equivariant lattice generating set, whose width (the number of variables necessary to write it down) depends linearly on the width of the map. This result is sharp and improves dramatically the previously known upper bound as it does not depend on the degree of the image monomial. In the case of of width two, we construct an explicit finite set of binomials generating the toric ideal up to symmetry. Both width and degree of this generating set are sharply bounded by linear functions in the exponents of the monomial.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"106 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122642929","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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