Let F= f1(A, X),...,fl(A, X) be a finite set of polynomials in Q[A, X] with variables A=A1,...,Am and X=X1,...,Xn. We study the continuity of the map θ from an element a of Cm to a subset of Cn defined by θ(a)= " the zeros of the polynomial ideal < f1(a, X),..., fl(a, X) >". Let G=(G1, S1),..., (Gk, Sk) be a comprehensive Gröbner system of < F > regarding A as parameters. By a basic property of a comprehensive Gröbner system, when the ideal < f1(a, X),..., fl(a, X) > is zero dimensional for some a ın Si, it is also zero dimensional for any a ın Si and the cardinality of θ(a) is identical on Si counting their multiplicities. In this paper, we prove that θ is also continuous on Si. Our result ensures the correctness of an algorithm for real quantifier elimination one of the authors has recently developed.
设F= f1(A, X)，…，fl(A, X)是Q[A, X]中多项式的有限集合，变量A=A1，…，Am and X=X1，…，Xn。我们研究了从Cm的元素a到Cn的子集的映射θ的连续性，其定义为θ(a)=“多项式理想< f1(a, X)，…， fl(a, X) > ' '。设G=(G1, S1)，…， (Gk, Sk)是以a为参数的< F >的综合Gröbner系统。利用综合Gröbner系统的基本性质，当理想< f1(a, X)时，…， fl(a, X) >对于某些a ın Si是零维的，对于任何a ın Si也是零维的θ(a)的基数在Si上是相同的，计算它们的多重性。本文证明了θ在Si上也是连续的。我们的结果保证了作者最近开发的一种消除实量词算法的正确性。
{"title":"On Continuity of the Roots of a Parametric Zero Dimensional Multivariate Polynomial Ideal","authors":"Yosuke Sato, Ryoya Fukasaku, Hiroshi Sekigawa","doi":"10.1145/3208976.3209004","DOIUrl":"https://doi.org/10.1145/3208976.3209004","url":null,"abstract":"Let F= f1(A, X),...,fl(A, X) be a finite set of polynomials in Q[A, X] with variables A=A1,...,Am and X=X1,...,Xn. We study the continuity of the map θ from an element a of Cm to a subset of Cn defined by θ(a)= \" the zeros of the polynomial ideal < f1(a, X),..., fl(a, X) >\". Let G=(G1, S1),..., (Gk, Sk) be a comprehensive Gröbner system of < F > regarding A as parameters. By a basic property of a comprehensive Gröbner system, when the ideal < f1(a, X),..., fl(a, X) > is zero dimensional for some a ın Si, it is also zero dimensional for any a ın Si and the cardinality of θ(a) is identical on Si counting their multiplicities. In this paper, we prove that θ is also continuous on Si. Our result ensures the correctness of an algorithm for real quantifier elimination one of the authors has recently developed.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132028732","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}
An algorithm is presented for computing the resultant of two generic bivariate polynomials over a field K. For such p and q K[x,y] both of degree d in x and n in y , the algorithm computes the resultant with respect to y using (n2 - 1/ømega d) 1+o(1) arithmetic operations in K, where two n x n matrices are multiplied using O(nømega) operations. Previous algorithms required time (n2 d) 1+o(1). The resultant is the determinant of the Sylvester matrix S(x) of p and q , which is an n x n Toeplitz-like polynomial matrix of degree~ d . We use a blocking technique and exploit the structure of S(x) for reducing the determinant computation to the computation of a matrix fraction description R(x)Q(x)-1 of an m x m submatrix of the inverse S(x)-1, where młl n. We rely on fast algorithms for handling dense polynomial matrices: the fraction description is obtained from an x -adic expansion via matrix fraction reconstruction, and the resultant as the determinant of the denominator matrix. We also describe some extensions of the approach to the computation of generic Gröbner bases and of characteristic polynomials of generic structured matrices and in univariate quotient algebras.
给出了一种计算域K上两个一般二元多项式的结式的算法。对于这样的p和q K[x,y]在x中都是d次，在y中都是n次，该算法在K中使用(n2 - 1/ømega d) 1+o(1)个算术运算来计算关于y的结式，其中两个n x n矩阵使用o(nømega)运算相乘。以前的算法需要时间(n2 d) 1+o(1)。结果是p和q的Sylvester矩阵S(x)的行列式，它是一个阶为~ d的n x n类toeplitz多项式矩阵。我们使用阻塞技术并利用S(x)的结构将行列式计算减少到矩阵分数描述R(x)Q(x)-1的逆S(x)-1的m x m子矩阵的计算，其中młl n。我们依赖于处理密集多项式矩阵的快速算法:分数描述是通过矩阵分数重建从x进展开获得的，结果作为分母矩阵的行列式。我们还描述了计算一般Gröbner基和一般结构矩阵的特征多项式和单变量商代数的方法的一些扩展。
{"title":"On Computing the Resultant of Generic Bivariate Polynomials","authors":"G. Villard","doi":"10.1145/3208976.3209020","DOIUrl":"https://doi.org/10.1145/3208976.3209020","url":null,"abstract":"An algorithm is presented for computing the resultant of two generic bivariate polynomials over a field K. For such p and q K[x,y] both of degree d in x and n in y , the algorithm computes the resultant with respect to y using (n2 - 1/ømega d) 1+o(1) arithmetic operations in K, where two n x n matrices are multiplied using O(nømega) operations. Previous algorithms required time (n2 d) 1+o(1). The resultant is the determinant of the Sylvester matrix S(x) of p and q , which is an n x n Toeplitz-like polynomial matrix of degree~ d . We use a blocking technique and exploit the structure of S(x) for reducing the determinant computation to the computation of a matrix fraction description R(x)Q(x)-1 of an m x m submatrix of the inverse S(x)-1, where młl n. We rely on fast algorithms for handling dense polynomial matrices: the fraction description is obtained from an x -adic expansion via matrix fraction reconstruction, and the resultant as the determinant of the denominator matrix. We also describe some extensions of the approach to the computation of generic Gröbner bases and of characteristic polynomials of generic structured matrices and in univariate quotient algebras.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"113 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124059463","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 use the method of characteristic sets with respect to two term orderings to prove the existence and obtain a method of computation of a bivariate dimension polynomial associated with a non-reflexive difference-differential ideal in the algebra of difference-differential polynomials with several basic derivations and one translation. As a consequence, we obtain a new proof and a method of computation of the dimension polynomial of a non-reflexive prime difference ideal in the algebra of difference polynomials over an ordinary difference field. We also discuss applications of our results to systems of algebraic difference-differential equations.
{"title":"Bivariate Dimension Polynomials of Non-Reflexive Prime Difference-Differential Ideals.: The Case of One Translation","authors":"A. Levin","doi":"10.1145/3208976.3209008","DOIUrl":"https://doi.org/10.1145/3208976.3209008","url":null,"abstract":"We use the method of characteristic sets with respect to two term orderings to prove the existence and obtain a method of computation of a bivariate dimension polynomial associated with a non-reflexive difference-differential ideal in the algebra of difference-differential polynomials with several basic derivations and one translation. As a consequence, we obtain a new proof and a method of computation of the dimension polynomial of a non-reflexive prime difference ideal in the algebra of difference polynomials over an ordinary difference field. We also discuss applications of our results to systems of algebraic difference-differential equations.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129711073","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 continue the investigations of the constructivity of arithmetics within non-commutative Ore localizations, initiated in our 2017 ISSAC paper, where we have introduced monoidal, geometric and rational types of localizations of domains as objects of our studies. Here we extend this classification to rings with zero divisors and consider Ore sets of the mentioned types which are commutative enough: such a set either belongs to a commutative algebra or it is central or its elements commute pairwise. By using the systematic approach we have developed before, we prove that arithmetic within the localization of a commutative polynomial algebra is constructive and give the necessary algorithms. We also address the important question of computing the local closure of ideals which is also known as the desingularization. We provide algorithms to compute such closures for certain non-commutative rings with respect to Ore sets with enough commutativity.
{"title":"Constructive Arithmetics in Ore Localizations with Enough Commutativity","authors":"Johannes Hoffmann, V. Levandovskyy","doi":"10.1145/3208976.3209021","DOIUrl":"https://doi.org/10.1145/3208976.3209021","url":null,"abstract":"We continue the investigations of the constructivity of arithmetics within non-commutative Ore localizations, initiated in our 2017 ISSAC paper, where we have introduced monoidal, geometric and rational types of localizations of domains as objects of our studies. Here we extend this classification to rings with zero divisors and consider Ore sets of the mentioned types which are commutative enough: such a set either belongs to a commutative algebra or it is central or its elements commute pairwise. By using the systematic approach we have developed before, we prove that arithmetic within the localization of a commutative polynomial algebra is constructive and give the necessary algorithms. We also address the important question of computing the local closure of ideals which is also known as the desingularization. We provide algorithms to compute such closures for certain non-commutative rings with respect to Ore sets with enough commutativity.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114156050","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 facet equations of a 3--dimensional alcoved polyhedron P are only of two types (xi=cnst and xi-xj=cnst) and the f --vector of P is bounded above by (20,30,12). In general, P is a dodecahedron with 20 vertices and 30 edges. We represent an alcoved polyhedron by a real square matrix A of order 4 and we compute the exact volume of P: it is a polynomial expression in the aij, homogeneous of degree 3 with rational coefficients. Then we compute the volume of the polar P o, when P is centrally symmetric. Last, we show that Mahler conjecture holds in this case: the product of the volumes of P and Po is no less that 43/3!, with equality only for boxes. Our proof reduces to computing a certificate of non--negativeness of a certain polynomial (in 3 variables, of degree 6, non homogeneous) on a certain simplex.
{"title":"Volume of Alcoved Polyhedra and Mahler Conjecture","authors":"M. J. Puente, Pedro L. Claveria","doi":"10.1145/3208976.3208990","DOIUrl":"https://doi.org/10.1145/3208976.3208990","url":null,"abstract":"The facet equations of a 3--dimensional alcoved polyhedron P are only of two types (xi=cnst and xi-xj=cnst) and the f --vector of P is bounded above by (20,30,12). In general, P is a dodecahedron with 20 vertices and 30 edges. We represent an alcoved polyhedron by a real square matrix A of order 4 and we compute the exact volume of P: it is a polynomial expression in the aij, homogeneous of degree 3 with rational coefficients. Then we compute the volume of the polar P o, when P is centrally symmetric. Last, we show that Mahler conjecture holds in this case: the product of the volumes of P and Po is no less that 43/3!, with equality only for boxes. Our proof reduces to computing a certificate of non--negativeness of a certain polynomial (in 3 variables, of degree 6, non homogeneous) on a certain simplex.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"57 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120810835","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 show that, for cyclic convolutional codes, it is possible to compute a sequence of positive integers, called cyclic column distances, which presents a more regular behavior than the classical column distances sequence. We then design an algorithm for the computation of the free distance based on the calculation of this cyclic column distances sequence.
{"title":"Computing Free Distances of Idempotent Convolutional Codes","authors":"J. Gómez-Torrecillas, F. J. Lobillo, G. Navarro","doi":"10.1145/3208976.3208985","DOIUrl":"https://doi.org/10.1145/3208976.3208985","url":null,"abstract":"We show that, for cyclic convolutional codes, it is possible to compute a sequence of positive integers, called cyclic column distances, which presents a more regular behavior than the classical column distances sequence. We then design an algorithm for the computation of the free distance based on the calculation of this cyclic column distances sequence.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133533519","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}
Let G be the reduced Grö bner basis of a zero-dimensional ideal I ⊆ K[X, Y] of bivariate polynomials over an effective field K. Modulo suitable regularity assumptions on G and suitable precomputations as a function of G , we prove the existence of a quasi-optimal algorithm for the reduction of polynomials in K [X, Y] with respect to G . Applications include fast algorithms for multiplication in the quotient algebra A=K[X, Y] / I and for conversions due to changes of the term ordering.
{"title":"Fast Reduction of Bivariate Polynomials with Respect to Sufficiently Regular Gröbner Bases","authors":"J. Hoeven, Robin Larrieu","doi":"10.1145/3208976.3209003","DOIUrl":"https://doi.org/10.1145/3208976.3209003","url":null,"abstract":"Let G be the reduced Grö bner basis of a zero-dimensional ideal I ⊆ K[X, Y] of bivariate polynomials over an effective field K. Modulo suitable regularity assumptions on G and suitable precomputations as a function of G , we prove the existence of a quasi-optimal algorithm for the reduction of polynomials in K [X, Y] with respect to G . Applications include fast algorithms for multiplication in the quotient algebra A=K[X, Y] / I and for conversions due to changes of the term ordering.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134461202","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}
Let f= (f1, ..., fs) be a sequence of polynomials in Q[X1,...,Xn] of maximal degree D and V⊂ Cn be the algebraic set defined by f and r be its dimension. The real radical re < f > associated to f is the largest ideal which defines the real trace of V . When V is smooth, we show that re < f >, has a finite set of generators with degrees bounded by V. Moreover, we present a probabilistic algorithm of complexity (snDn )O(1) to compute the minimal primes of re < f >. When V is not smooth, we give a probabilistic algorithm of complexity sO(1) (nD)O(nr2r) to compute rational parametrizations for all irreducible components of the real algebraic set V ∩ Rn. Experiments are given to show the efficiency of our approaches.
令f= (f1，…)， fs)是Q[X1，…]中的多项式序列。，Xn]最大次D, V∧Cn是由f定义的代数集，r是其维数。与f相关的实数根re < f >是定义V的实数迹的最大理想。当V光滑时，我们证明了re < f >有一个有限的生成器集合，它们的度以V为界。此外，我们给出了一个复杂度为(snDn)O(1)的概率算法来计算re < f >的最小素数。当V不光滑时，我们给出了一个复杂度为sO(1) (nD)O(nr2r)的概率算法来计算实代数集合V∩Rn的所有不可约分量的有理参数化。实验表明了所提方法的有效性。
{"title":"On the Complexity of Computing Real Radicals of Polynomial Systems","authors":"M. S. E. Din, Zhi-Hong Yang, L. Zhi","doi":"10.1145/3208976.3209002","DOIUrl":"https://doi.org/10.1145/3208976.3209002","url":null,"abstract":"Let f= (f1, ..., fs) be a sequence of polynomials in Q[X1,...,Xn] of maximal degree D and V⊂ Cn be the algebraic set defined by f and r be its dimension. The real radical re < f > associated to f is the largest ideal which defines the real trace of V . When V is smooth, we show that re < f >, has a finite set of generators with degrees bounded by V. Moreover, we present a probabilistic algorithm of complexity (snDn )O(1) to compute the minimal primes of re < f >. When V is not smooth, we give a probabilistic algorithm of complexity sO(1) (nD)O(nr2r) to compute rational parametrizations for all irreducible components of the real algebraic set V ∩ Rn. Experiments are given to show the efficiency of our approaches.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127192429","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}
Many special functions as well as generating functions of combinatorial sequences that arise in applications are D-finite, i.e., they satisfy a linear differential equation with polynomial coefficients. These functions have been studied for centuries and over the past decades various computer algebra methods have been developed and implemented for D-finite functions. Recently, we have extended this notion to DD-finite functions (functions satisfying linear differential equations with D-finite functions coefficients). Numerous identities for D-finite functions can be proven automatically using closure properties. These closure properties can be shown to hold for DD-finite functions as well. In this paper, we present the algorithmic aspect of these closure properties, discuss issues related to implementation and give several examples.
{"title":"Algorithmic Arithmetics with DD-Finite Functions","authors":"Antonio Jiménez-Pastor, V. Pillwein","doi":"10.1145/3208976.3209009","DOIUrl":"https://doi.org/10.1145/3208976.3209009","url":null,"abstract":"Many special functions as well as generating functions of combinatorial sequences that arise in applications are D-finite, i.e., they satisfy a linear differential equation with polynomial coefficients. These functions have been studied for centuries and over the past decades various computer algebra methods have been developed and implemented for D-finite functions. Recently, we have extended this notion to DD-finite functions (functions satisfying linear differential equations with D-finite functions coefficients). Numerous identities for D-finite functions can be proven automatically using closure properties. These closure properties can be shown to hold for DD-finite functions as well. In this paper, we present the algorithmic aspect of these closure properties, discuss issues related to implementation and give several examples.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125316101","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}
Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp--Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidimensional cyclic code decoding require guessing linear recurrence relations of a multivariate sequence. Several algorithms solve this problem. The so-called Berlekamp--Massey--Sakata algorithm (1988) uses polynomial additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on linear algebra operations on a multi-Hankel matrix, a multivariate generalization of a Hankel matrix. The Artinian Gorenstein border basis algorithm (2017) uses a Gram-Schmidt process. We propose a new algorithm for computing the Gröbner basis of the ideal of relations of a sequence based solely on multivariate polynomial arithmetic. This algorithm allows us to both revisit the Berlekamp--Massey--Sakata algorithm through the use of polynomial divisions and to completely revise the Scalar-FGLM algorithm without linear algebra operations. A key observation in the design of this algorithm is to work on the mirror of the truncated generating series allowing us to use polynomial arithmetic modulo a monomial ideal. It appears to have some similarities with Padé approximants of this mirror polynomial. Finally, we give a partial solution to the transformation of this algorithm into an adaptive one.
{"title":"A Polynomial-Division-Based Algorithm for Computing Linear Recurrence Relations","authors":"Jérémy Berthomieu, J. Faugère","doi":"10.1145/3208976.3209017","DOIUrl":"https://doi.org/10.1145/3208976.3209017","url":null,"abstract":"Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp--Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidimensional cyclic code decoding require guessing linear recurrence relations of a multivariate sequence. Several algorithms solve this problem. The so-called Berlekamp--Massey--Sakata algorithm (1988) uses polynomial additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on linear algebra operations on a multi-Hankel matrix, a multivariate generalization of a Hankel matrix. The Artinian Gorenstein border basis algorithm (2017) uses a Gram-Schmidt process. We propose a new algorithm for computing the Gröbner basis of the ideal of relations of a sequence based solely on multivariate polynomial arithmetic. This algorithm allows us to both revisit the Berlekamp--Massey--Sakata algorithm through the use of polynomial divisions and to completely revise the Scalar-FGLM algorithm without linear algebra operations. A key observation in the design of this algorithm is to work on the mirror of the truncated generating series allowing us to use polynomial arithmetic modulo a monomial ideal. It appears to have some similarities with Padé approximants of this mirror polynomial. Finally, we give a partial solution to the transformation of this algorithm into an adaptive one.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"201 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123205465","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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