Consider a matrix F ε K [x]^mxn of univariate polynomials over a field K. We study the problem of computing the column rank profile of F. To this end we first give an algorithm which improves the minimal kernel basis algorithm of Zhou, Labahn, and Storjohann (Proceedings ISSAC 2012). We then provide a second algorithm which computes the column rank profile of F with a rank-sensitive complexity of O~ (rw-2n(m+d)) operations in K. Here, D is the sum of row degrees of F, w is the exponent of matrix multiplication, and O~ (.) hides logarithmic factors.
考虑域K上的单变量多项式的矩阵F ε K [x]^mxn。我们研究了F的列秩轮廓的计算问题。为此，我们首先给出了一种算法，该算法改进了Zhou, Labahn和Storjohann的最小核基算法(Proceedings ISSAC 2012)。然后，我们提供了第二种算法，该算法计算F的列秩轮廓，其秩敏感复杂度为k中的O~ (rw-2n(m+d))次操作。这里，d是F的行度和，w是矩阵乘法的指数，O~(.)隐藏对数因子。
{"title":"Rank-Sensitive Computation of the Rank Profile of a Polynomial Matrix","authors":"G. Labahn, Vincent Neiger, Thi Xuan Vu, Wei Zhou","doi":"10.1145/3476446.3535495","DOIUrl":"https://doi.org/10.1145/3476446.3535495","url":null,"abstract":"Consider a matrix F ε K [x]^mxn of univariate polynomials over a field K. We study the problem of computing the column rank profile of F. To this end we first give an algorithm which improves the minimal kernel basis algorithm of Zhou, Labahn, and Storjohann (Proceedings ISSAC 2012). We then provide a second algorithm which computes the column rank profile of F with a rank-sensitive complexity of O~ (rw-2n(m+d)) operations in K. Here, D is the sum of row degrees of F, w is the exponent of matrix multiplication, and O~ (.) hides logarithmic factors.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128278269","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}
Linear recurrence operators in characteristic p are classified by their p-curvature. For a recurrence operator L, denote by χ(L) the characteristic polynomial of its p-curvature. We can obtain information about the factorization of L by factoring χ(L). The main theorem of this paper gives an unexpected relation between χ(L) and the true singularities of L. An application is to speed up a fast algorithm for computing χ(L) by desingularizing L first. Another contribution of this paper is faster desingularization.
{"title":"Desingularization and p-Curvature of Recurrence Operators","authors":"Yi Zhou, M. V. Hoeij","doi":"10.1145/3476446.3535478","DOIUrl":"https://doi.org/10.1145/3476446.3535478","url":null,"abstract":"Linear recurrence operators in characteristic p are classified by their p-curvature. For a recurrence operator L, denote by χ(L) the characteristic polynomial of its p-curvature. We can obtain information about the factorization of L by factoring χ(L). The main theorem of this paper gives an unexpected relation between χ(L) and the true singularities of L. An application is to speed up a fast algorithm for computing χ(L) by desingularizing L first. Another contribution of this paper is faster desingularization.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"12 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124185312","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}
Reconstructing a hypothetical recurrence equation from the first terms of an infinite sequence is a classical and well-known technique in experimental mathematics. We propose a variation of this technique which can succeed with fewer input terms.
{"title":"Guessing with Little Data","authors":"Manuel Kauers, C. Koutschan","doi":"10.1145/3476446.3535486","DOIUrl":"https://doi.org/10.1145/3476446.3535486","url":null,"abstract":"Reconstructing a hypothetical recurrence equation from the first terms of an infinite sequence is a classical and well-known technique in experimental mathematics. We propose a variation of this technique which can succeed with fewer input terms.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114611823","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}
Pascal Giorgi, Bruno Grenet, Armelle Perret du Cray, Daniel S. Roche
Given a way to evaluate an unknown polynomial with integer coefficients, we present new algorithms to recover its nonzero coefficients and corresponding exponents. As an application, we adapt this interpolation algorithm to the problem of computing the exact quotient of two given polynomials. These methods are efficient in terms of the bit-length of the sparse representation, that is, the number of nonzero terms, the size of coefficients, the number of variables, and the logarithm of the degree. At the core of our results is a new Monte Carlo randomized algorithm to recover a polynomial f(x) with integer coefficients given a way to evaluate f(θ) mod m for any chosen integers θ and m. This algorithm has nearly-optimal bit complexity, meaning that the total bit-length of the probes, as well as the computational running time, is softly linear (ignoring logarithmic factors) in the bit-length of the resulting sparse polynomial. To our knowledge, this is the first sparse interpolation algorithm with soft-linear bit complexity in the total output size. For polynomials with integer coefficients, the best previously known results have at least a cubic dependency on the bit-length of the exponents.
给出了一种求整数系数的未知多项式的方法，给出了恢复其非零系数和相应指数的新算法。作为一个应用，我们将该插值算法应用于计算两个给定多项式的精确商的问题。这些方法在稀疏表示的位长度方面是有效的，即非零项的数量、系数的大小、变量的数量和度的对数。在我们的结果的核心是一个新的蒙特卡罗随机算法，以恢复多项式f(x)与整数系数给定一种方法来评估f(θ) mod m对任何选择的整数θ和m。该算法具有近乎最优的位复杂度，这意味着探针的总位长度，以及计算运行时间，是软线性(忽略对数因素)在得到的稀疏多项式的位长度。据我们所知，这是第一个在总输出大小中具有软线性位复杂度的稀疏插值算法。对于整数系数的多项式，最好的已知结果与指数的位长度至少有三次依赖关系。
{"title":"Sparse Polynomial Interpolation and Division in Soft-linear Time","authors":"Pascal Giorgi, Bruno Grenet, Armelle Perret du Cray, Daniel S. Roche","doi":"10.1145/3476446.3536173","DOIUrl":"https://doi.org/10.1145/3476446.3536173","url":null,"abstract":"Given a way to evaluate an unknown polynomial with integer coefficients, we present new algorithms to recover its nonzero coefficients and corresponding exponents. As an application, we adapt this interpolation algorithm to the problem of computing the exact quotient of two given polynomials. These methods are efficient in terms of the bit-length of the sparse representation, that is, the number of nonzero terms, the size of coefficients, the number of variables, and the logarithm of the degree. At the core of our results is a new Monte Carlo randomized algorithm to recover a polynomial f(x) with integer coefficients given a way to evaluate f(θ) mod m for any chosen integers θ and m. This algorithm has nearly-optimal bit complexity, meaning that the total bit-length of the probes, as well as the computational running time, is softly linear (ignoring logarithmic factors) in the bit-length of the resulting sparse polynomial. To our knowledge, this is the first sparse interpolation algorithm with soft-linear bit complexity in the total output size. For polynomials with integer coefficients, the best previously known results have at least a cubic dependency on the bit-length of the exponents.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115232084","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}
Klara Nosan, Amaury Pouly, M. Shirmohammadi, J. Worrell
We investigate the Membership Problem for hypergeometric sequences: given a hypergeometric sequence ❬un❭∞n=0 of rational numbers and a target t∈Q, decide whether t occurs in the sequence. We show decidability of this problem under the assumption that in the defining recurrence p(n)un = q(n)un-1, the roots of the polynomials p(x) and q(x) are all rational numbers. Our proof relies on bounds on the density of primes in arithmetic progressions. We also observe a relationship between the decidability of the Membership problem (and variants) and the Rohrlich-Lang conjecture in transcendence theory.
{"title":"The Membership Problem for Hypergeometric Sequences with Rational Parameters","authors":"Klara Nosan, Amaury Pouly, M. Shirmohammadi, J. Worrell","doi":"10.1145/3476446.3535504","DOIUrl":"https://doi.org/10.1145/3476446.3535504","url":null,"abstract":"We investigate the Membership Problem for hypergeometric sequences: given a hypergeometric sequence ❬un❭∞n=0 of rational numbers and a target t∈Q, decide whether t occurs in the sequence. We show decidability of this problem under the assumption that in the defining recurrence p(n)un = q(n)un-1, the roots of the polynomials p(x) and q(x) are all rational numbers. Our proof relies on bounds on the density of primes in arithmetic progressions. We also observe a relationship between the decidability of the Membership problem (and variants) and the Rohrlich-Lang conjecture in transcendence theory.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123774283","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}
Robert M Corless, G. Labahn, Dan Piponi, Leili Rafiee Sevyeri
A Bohemian matrix family is a set of matrices all of whose entries are drawn from a fixed, usually discrete and hence bounded, subset of a field of characteristic zero. Originally these were integers---hence the name, from the acronym BOunded HEight Matrix of Integers (BOHEMI)---but other kinds of entries are also interesting. Some kinds of questions about Bohemian matrices can be answered by numerical computation, but sometimes exact computation is better. In this paper we explore some Bohemian families (symmetric, upper Hessenberg, or Toeplitz) computationally, and answer some open questions posed about the distributions of eigenvalue densities.
{"title":"Bohemian Matrix Geometry","authors":"Robert M Corless, G. Labahn, Dan Piponi, Leili Rafiee Sevyeri","doi":"10.1145/3476446.3536177","DOIUrl":"https://doi.org/10.1145/3476446.3536177","url":null,"abstract":"A Bohemian matrix family is a set of matrices all of whose entries are drawn from a fixed, usually discrete and hence bounded, subset of a field of characteristic zero. Originally these were integers---hence the name, from the acronym BOunded HEight Matrix of Integers (BOHEMI)---but other kinds of entries are also interesting. Some kinds of questions about Bohemian matrices can be answered by numerical computation, but sometimes exact computation is better. In this paper we explore some Bohemian families (symmetric, upper Hessenberg, or Toeplitz) computationally, and answer some open questions posed about the distributions of eigenvalue densities.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122937605","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 study ideals spanned by polynomials or overconvergent series in a Tate algebra. With state-of-the-art algorithms for computing Tate Gröbner bases, even if the input is polynomials, the size of the output grows with the required precision, both in terms of the size of the coefficients and the size of the support of the series. We prove that ideals which are spanned by polynomials admit a Tate Gröbner basis made of polynomials, and we propose an algorithm, leveraging Mora's weak normal form algorithm, for computing it. As a result, the size of the output of this algorithm grows linearly with the precision. Following the same ideas, we propose an algorithm which computes an overconvergent basis for an ideal spanned by overconvergent series. Finally, we prove the existence of a universal analytic Gröbner basis for polynomial ideals in Tate algebras, compatible with all convergence radii.
{"title":"On Polynomial Ideals and Overconvergence in Tate Algebras","authors":"X. Caruso, Tristan Vaccon, Thibaut Verron","doi":"10.1145/3476446.3535491","DOIUrl":"https://doi.org/10.1145/3476446.3535491","url":null,"abstract":"In this paper, we study ideals spanned by polynomials or overconvergent series in a Tate algebra. With state-of-the-art algorithms for computing Tate Gröbner bases, even if the input is polynomials, the size of the output grows with the required precision, both in terms of the size of the coefficients and the size of the support of the series. We prove that ideals which are spanned by polynomials admit a Tate Gröbner basis made of polynomials, and we propose an algorithm, leveraging Mora's weak normal form algorithm, for computing it. As a result, the size of the output of this algorithm grows linearly with the precision. Following the same ideas, we propose an algorithm which computes an overconvergent basis for an ideal spanned by overconvergent series. Finally, we prove the existence of a universal analytic Gröbner basis for polynomial ideals in Tate algebras, compatible with all convergence radii.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122395288","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}
Victor Magron, M. S. E. Din, M. Schweighofer, T. Vu
Certifying the positivity of trigonometric polynomials is of first importance for design problems in discrete-time signal processing. It is well known from the Riesz-Fejér spectral factorization theorem that any trigonometric univariate polynomial non-negative on the unit circle can be decomposed as a Hermitian square with complex coefficients. Here we focus on the case of polynomials with Gaussian integer coefficients, i.e., with real and imaginary parts being integers. We design, analyze and compare, theoretically and practically, three hybrid numeric-symbolic algorithms computing weighted sums of Hermitian squares decompositions for trigonometric univariate polynomials positive on the unit circle with Gaussian coefficients. The numerical steps the first and second algorithm rely on are complex root isolation and semidefinite programming, respectively. An exact sum of Hermitian squares decomposition is obtained thanks to compensation techniques. The third algorithm, also based on complex semidefinite programming, is an adaptation of the rounding and projection algorithm by Peyrl and Parrilo. For all three algorithms, we prove bit complexity and output size estimates that are polynomial in the degree of the input and linear in the maximum bitsize of its coefficients. We compare their performance on randomly chosen benchmarks, and further design a certified finite impulse filter.
{"title":"Exact SOHS Decompositions of Trigonometric Univariate Polynomials with Gaussian Coefficients","authors":"Victor Magron, M. S. E. Din, M. Schweighofer, T. Vu","doi":"10.1145/3476446.3535480","DOIUrl":"https://doi.org/10.1145/3476446.3535480","url":null,"abstract":"Certifying the positivity of trigonometric polynomials is of first importance for design problems in discrete-time signal processing. It is well known from the Riesz-Fejér spectral factorization theorem that any trigonometric univariate polynomial non-negative on the unit circle can be decomposed as a Hermitian square with complex coefficients. Here we focus on the case of polynomials with Gaussian integer coefficients, i.e., with real and imaginary parts being integers. We design, analyze and compare, theoretically and practically, three hybrid numeric-symbolic algorithms computing weighted sums of Hermitian squares decompositions for trigonometric univariate polynomials positive on the unit circle with Gaussian coefficients. The numerical steps the first and second algorithm rely on are complex root isolation and semidefinite programming, respectively. An exact sum of Hermitian squares decomposition is obtained thanks to compensation techniques. The third algorithm, also based on complex semidefinite programming, is an adaptation of the rounding and projection algorithm by Peyrl and Parrilo. For all three algorithms, we prove bit complexity and output size estimates that are polynomial in the degree of the input and linear in the maximum bitsize of its coefficients. We compare their performance on randomly chosen benchmarks, and further design a certified finite impulse filter.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"10 16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132141847","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. Ergur, Josué Tonelli-Cueto, Elias P. Tsigaridas
Isolating the real roots of univariate polynomials is a fundamental problem in symbolic computation and it is arguably one of the most important problems in computational mathematics. The problem has a long history decorated with numerous ingenious algorithms and furnishes an active area of research. However, the worst-case analysis of root-finding algorithms does not correlate with their practical performance. We develop a smoothed analysis framework for polynomials with integer coefficients to bridge the gap between the complexity estimates and the practical performance. In this setting, we derive that the expected bit complexity of Descartes solver to isolate the real roots of a polynomial, with coefficients uniformly distributed, is ÕB(d2 + dτ), where d is the degree of the polynomial and τ the bitsize of the coefficients.
{"title":"Beyond Worst-Case Analysis for Root Isolation Algorithms","authors":"A. Ergur, Josué Tonelli-Cueto, Elias P. Tsigaridas","doi":"10.1145/3476446.3535475","DOIUrl":"https://doi.org/10.1145/3476446.3535475","url":null,"abstract":"Isolating the real roots of univariate polynomials is a fundamental problem in symbolic computation and it is arguably one of the most important problems in computational mathematics. The problem has a long history decorated with numerous ingenious algorithms and furnishes an active area of research. However, the worst-case analysis of root-finding algorithms does not correlate with their practical performance. We develop a smoothed analysis framework for polynomials with integer coefficients to bridge the gap between the complexity estimates and the practical performance. In this setting, we derive that the expected bit complexity of Descartes solver to isolate the real roots of a polynomial, with coefficients uniformly distributed, is ÕB(d2 + dτ), where d is the degree of the polynomial and τ the bitsize of the coefficients.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"113 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132810430","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 aims at initializing a dynamical aspect of symbolic integration by studying stability problems in differential fields. We first show some basic properties of stable elementary functions and then characterize three special families of stable elementary functions including rational functions, logarithmic functions, and exponential functions. We prove that all D-finite power series are eventually stable. Some problems for future studies are proposed towards deeper dynamical studies in differential algebra.
{"title":"Stability Problems in Symbolic Integration","authors":"Shaoshi Chen","doi":"10.1145/3476446.3535502","DOIUrl":"https://doi.org/10.1145/3476446.3535502","url":null,"abstract":"This paper aims at initializing a dynamical aspect of symbolic integration by studying stability problems in differential fields. We first show some basic properties of stable elementary functions and then characterize three special families of stable elementary functions including rational functions, logarithmic functions, and exponential functions. We prove that all D-finite power series are eventually stable. Some problems for future studies are proposed towards deeper dynamical studies in differential algebra.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114182147","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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