Pub Date : 2024-10-11DOI: 10.1016/j.jsc.2024.102395
Jane Ivy Coons , Maize Curiel , Elizabeth Gross
The steady-state degree of a chemical reaction network is the number of complex steady-states for generic rate constants and initial conditions. One way to bound the steady-state degree is through the mixed volume of an associated steady-state system. In this work, we show that for partitionable binomial chemical reaction systems, whose resulting steady-state systems are given by a set of binomials and a set of linear (not necessarily binomial) conservation equations, computing the mixed volume is equivalent to finding the volume of a single mixed cell that is the translate of a parallelotope. Additionally, we give a coloring condition on cycle networks to identify reaction systems with binomial steady-state ideals. We highlight both of these theorems using a class of networks referred to as species-overlapping networks and give a formula for the mixed volume of these networks.
{"title":"Mixed volumes of networks with binomial steady-states","authors":"Jane Ivy Coons , Maize Curiel , Elizabeth Gross","doi":"10.1016/j.jsc.2024.102395","DOIUrl":"10.1016/j.jsc.2024.102395","url":null,"abstract":"<div><div>The steady-state degree of a chemical reaction network is the number of complex steady-states for generic rate constants and initial conditions. One way to bound the steady-state degree is through the mixed volume of an associated steady-state system. In this work, we show that for partitionable binomial chemical reaction systems, whose resulting steady-state systems are given by a set of binomials and a set of linear (not necessarily binomial) conservation equations, computing the mixed volume is equivalent to finding the volume of a single mixed cell that is the translate of a parallelotope. Additionally, we give a coloring condition on cycle networks to identify reaction systems with binomial steady-state ideals. We highlight both of these theorems using a class of networks referred to as species-overlapping networks and give a formula for the mixed volume of these networks.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102395"},"PeriodicalIF":0.6,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142532191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We determine the number of complex solutions to a nonlinear eigenvalue problem on the Grassmannian in its Plücker embedding. This is motivated by quantum chemistry, where it represents the truncation to single electrons in coupled cluster theory. We prove the formula for the Grassmannian of lines which was conjectured in earlier work with Fabian Faulstich. This rests on the geometry of the graph of a birational parametrization of the Grassmannian. We present a squarefree Gröbner basis for this graph, and we develop connections to toric degenerations from representation theory.
{"title":"Coupled cluster degree of the Grassmannian","authors":"Viktoriia Borovik , Bernd Sturmfels , Svala Sverrisdóttir","doi":"10.1016/j.jsc.2024.102396","DOIUrl":"10.1016/j.jsc.2024.102396","url":null,"abstract":"<div><div>We determine the number of complex solutions to a nonlinear eigenvalue problem on the Grassmannian in its Plücker embedding. This is motivated by quantum chemistry, where it represents the truncation to single electrons in coupled cluster theory. We prove the formula for the Grassmannian of lines which was conjectured in earlier work with Fabian Faulstich. This rests on the geometry of the graph of a birational parametrization of the Grassmannian. We present a squarefree Gröbner basis for this graph, and we develop connections to toric degenerations from representation theory.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102396"},"PeriodicalIF":0.6,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142437932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-10DOI: 10.1016/j.jsc.2024.102394
Peter Paule , Carsten Schneider
We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which guarantees the applicability of our method for many input sums. In addition, we elaborate new techniques to optimize the underlying key task of our method to compute rational solutions of parameterized linear recurrences.
{"title":"Creative telescoping for hypergeometric double sums","authors":"Peter Paule , Carsten Schneider","doi":"10.1016/j.jsc.2024.102394","DOIUrl":"10.1016/j.jsc.2024.102394","url":null,"abstract":"<div><div>We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which guarantees the applicability of our method for many input sums. In addition, we elaborate new techniques to optimize the underlying key task of our method to compute rational solutions of parameterized linear recurrences.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102394"},"PeriodicalIF":0.6,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142437933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The equivariant nonnegativity versus sums of squares question has been solved for any infinite series of essential reflection groups but type A. As a first step to a classification, we analyse -invariant quartics. We prove that the cones of invariant sums of squares and nonnegative forms are equal if and only if the number of variables is at most 3 or odd.
对于除 A 型之外的任何无限序列本质反射群，等变非负性与平方和问题都已解决。我们证明，当且仅当变量数至多为 3 或奇数时，不变平方和与非负形式的锥相等。
{"title":"On nonnegative invariant quartics in type A","authors":"Sebastian Debus , Charu Goel , Salma Kuhlmann , Cordian Riener","doi":"10.1016/j.jsc.2024.102393","DOIUrl":"10.1016/j.jsc.2024.102393","url":null,"abstract":"<div><div>The equivariant nonnegativity versus sums of squares question has been solved for any infinite series of essential reflection groups but type <em>A</em>. As a first step to a classification, we analyse <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-invariant quartics. We prove that the cones of invariant sums of squares and nonnegative forms are equal if and only if the number of variables is at most 3 or odd.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102393"},"PeriodicalIF":0.6,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142433151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-05DOI: 10.1016/j.jsc.2024.102388
Zijia Li , Hans-Peter Schröcker , Johannes Siegele
Rational motions in conformal three space can be parametrized by polynomials with coefficients in a suitable Clifford algebra. We call them “spinor polynomials.” In this text we present a new algorithm to decompose generic spinor polynomials into linear factors. The factorization algorithm is based on the “kinematics at infinity”. Factorizations exist generically but not generally and are typically not unique. We prove that generic multiples of non-factorizable spinor polynomials admit factorizations and we demonstrate at hand of an example how our ideas can be used to tackle the hitherto unsolved problem of “factorizing” algebraic motions.
{"title":"A geometric algorithm for the factorization of rational motions in conformal three space","authors":"Zijia Li , Hans-Peter Schröcker , Johannes Siegele","doi":"10.1016/j.jsc.2024.102388","DOIUrl":"10.1016/j.jsc.2024.102388","url":null,"abstract":"<div><div>Rational motions in conformal three space can be parametrized by polynomials with coefficients in a suitable Clifford algebra. We call them “spinor polynomials.” In this text we present a new algorithm to decompose generic spinor polynomials into linear factors. The factorization algorithm is based on the “kinematics at infinity”. Factorizations exist generically but not generally and are typically not unique. We prove that generic multiples of non-factorizable spinor polynomials admit factorizations and we demonstrate at hand of an example how our ideas can be used to tackle the hitherto unsolved problem of “factorizing” algebraic motions.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102388"},"PeriodicalIF":0.6,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142532095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-04DOI: 10.1016/j.jsc.2024.102391
Amir Hashemi , Deepak Kapur
A new approach for Gröbner bases conversion of polynomial ideals (over a field) of arbitrary dimension is presented. In contrast to the only other approach based on Gröbner fan and Gröbner walk for positive dimensional ideals, the proposed approach is simpler to understand, prove, and implement. It is based on defining for a given polynomial, a truncated sub-polynomial consisting of all monomials that can possibly become the leading monomial with respect to the target ordering: the monomials between the leading monomial of the target ordering and the leading monomial of the initial ordering.
The main ingredient of the new algorithm is the computation of a Gröbner basis with respect to the target ordering for the ideal generated by such truncated parts of the input Gröbner basis. This is done using the extended Buchberger algorithm that also outputs the relationship between the input and output bases. That information is used in attempts to recover a Gröbner basis of the ideal with respect to the target ordering. In general, more than one iteration may be needed to get a Gröbner basis with respect to the target ordering since truncated polynomials may miss some leading monomials.
The new algorithm has been implemented in Maple and its operation is illustrated using an example. The performance of this implementation is compared with the implementations of other approaches in Maple. In practice, a Gröbner basis with respect to a target ordering can be computed in a single iteration on most examples.
Since the proposed basis conversion algorithm uses simple concepts of Gröbner basis theory, it can be easily taught in contrast to methods based on Gröbner walk.
本文提出了一种转换任意维度多项式理想（在一个域上）的格罗布纳基的新方法。与其他唯一基于格罗伯纳扇形和格罗伯纳走正维理想的方法相比，所提出的方法更易于理解、证明和实施。它的基础是为给定的多项式定义一个截断的子多项式，该子多项式由所有可能成为目标排序的前导单项式的单项式组成：目标排序的前导单项式和初始排序的前导单项式之间的单项式。计算是通过扩展的布赫伯格算法完成的，该算法还能输出输入和输出基础之间的关系。这些信息将被用于恢复与目标排序相关的理想格罗伯纳基。一般来说，由于截断多项式可能会遗漏一些前导单项式，因此可能需要不止一次迭代才能得到与目标排序相关的格罗伯纳基。新算法已在 Maple 中实现，并通过一个例子对其操作进行了说明。该实现方法的性能与 Maple 中其他方法的实现方法进行了比较。实际上，在大多数例子中，一次迭代就可以计算出与目标排序相关的格罗伯纳基础。由于所提出的基础转换算法使用的是格罗伯纳基础理论的简单概念，因此与基于格罗伯纳行走的方法相比，它很容易教授。
{"title":"A new algorithm for Gröbner bases conversion","authors":"Amir Hashemi , Deepak Kapur","doi":"10.1016/j.jsc.2024.102391","DOIUrl":"10.1016/j.jsc.2024.102391","url":null,"abstract":"<div><div>A new approach for Gröbner bases conversion of polynomial ideals (over a field) of arbitrary dimension is presented. In contrast to the only other approach based on Gröbner fan and Gröbner walk for positive dimensional ideals, the proposed approach is simpler to understand, prove, and implement. It is based on defining for a given polynomial, a truncated sub-polynomial consisting of all monomials that can possibly become the leading monomial with respect to the target ordering: the monomials between the leading monomial of the target ordering and the leading monomial of the initial ordering.</div><div>The main ingredient of the new algorithm is the computation of a Gröbner basis with respect to the target ordering for the ideal generated by such truncated parts of the input Gröbner basis. This is done using the extended Buchberger algorithm that also outputs the relationship between the input and output bases. That information is used in attempts to recover a Gröbner basis of the ideal with respect to the target ordering. In general, more than one iteration may be needed to get a Gröbner basis with respect to the target ordering since truncated polynomials may miss some leading monomials.</div><div>The new algorithm has been implemented in <span>Maple</span> and its operation is illustrated using an example. The performance of this implementation is compared with the implementations of other approaches in <span>Maple.</span> In practice, a Gröbner basis with respect to a target ordering can be computed in a single iteration on most examples.</div><div>Since the proposed basis conversion algorithm uses simple concepts of Gröbner basis theory, it can be easily taught in contrast to methods based on Gröbner walk.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102391"},"PeriodicalIF":0.6,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-03DOI: 10.1016/j.jsc.2024.102392
Erik Parkinson , Kate Wall , Jane Slagle , Daniel Treuhaft , Xander de la Bruere , Samuel Goldrup , Timothy Keith , Peter Call , Tyler J. Jarvis
We present a new algorithm for finding isolated zeros of a system of real-valued functions in a bounded interval in . It uses the Chebyshev proxy method combined with a mixture of subdivision, reduction methods, and elimination checks that leverage special properties of Chebyshev polynomials. We prove the method has quadratic convergence locally near simple zeros of the system. It also finds all nonsimple zeros, but convergence to those zeros is not guaranteed to be quadratic. We also analyze the arithmetic complexity and the numerical stability of the algorithm and provide numerical evidence in dimensions up to five that the method is both fast and accurate on a wide range of problems. Our tests show that the algorithm outperforms other standard methods on the problem of finding all real zeros in a bounded domain. Our Python implementation of the algorithm is publicly available at https://github.com/tylerjarvis/RootFinding.
{"title":"Chebyshev subdivision and reduction methods for solving multivariable systems of equations","authors":"Erik Parkinson , Kate Wall , Jane Slagle , Daniel Treuhaft , Xander de la Bruere , Samuel Goldrup , Timothy Keith , Peter Call , Tyler J. Jarvis","doi":"10.1016/j.jsc.2024.102392","DOIUrl":"10.1016/j.jsc.2024.102392","url":null,"abstract":"<div><div>We present a new algorithm for finding isolated zeros of a system of real-valued functions in a bounded interval in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. It uses the Chebyshev proxy method combined with a mixture of subdivision, reduction methods, and elimination checks that leverage special properties of Chebyshev polynomials. We prove the method has quadratic convergence locally near simple zeros of the system. It also finds all nonsimple zeros, but convergence to those zeros is not guaranteed to be quadratic. We also analyze the arithmetic complexity and the numerical stability of the algorithm and provide numerical evidence in dimensions up to five that the method is both fast and accurate on a wide range of problems. Our tests show that the algorithm outperforms other standard methods on the problem of finding all real zeros in a bounded domain. Our Python implementation of the algorithm is publicly available at <span><span>https://github.com/tylerjarvis/RootFinding</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102392"},"PeriodicalIF":0.6,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142422636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-27DOI: 10.1016/j.jsc.2024.102390
Niels Lubbes
We classify the singular loci of real surfaces in three-space that contain two circles through each point. We characterize how a circle in such a surface meets this loci as it moves in its pencil and as such provide insight into the topology of the surface.
{"title":"Self-intersections of surfaces that contain two circles through each point","authors":"Niels Lubbes","doi":"10.1016/j.jsc.2024.102390","DOIUrl":"10.1016/j.jsc.2024.102390","url":null,"abstract":"<div><div>We classify the singular loci of real surfaces in three-space that contain two circles through each point. We characterize how a circle in such a surface meets this loci as it moves in its pencil and as such provide insight into the topology of the surface.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102390"},"PeriodicalIF":0.6,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142422634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-26DOI: 10.1016/j.jsc.2024.102386
Alexey A. Kytmanov , Sergey P. Tsarev
In this paper we prove that classical discrete orthogonal polynomials (Hahn polynomials on an equidistant grid with unit weights) of high degrees have extremely small values near the endpoints (we call this property “rapid decay near the endpoints”) but extremely large values between these grid points and their roots are very close to the grid points near the endpoints. These results imply important general boundary effects for stable linear polynomial filters (we call this property “rapid boundary attenuation”).
Our results give interesting examples of nontrivial asymptotics of practically important solutions of special second-order linear recurrencies with polynomial coefficients studied by M.Petkovšek; to his memory we dedicate this paper.
{"title":"Asymptotics of solutions of special second-order linear recurrencies with polynomial coefficients and boundary effects of polynomial filters","authors":"Alexey A. Kytmanov , Sergey P. Tsarev","doi":"10.1016/j.jsc.2024.102386","DOIUrl":"10.1016/j.jsc.2024.102386","url":null,"abstract":"<div><div>In this paper we prove that classical discrete orthogonal polynomials (Hahn polynomials on an equidistant grid with unit weights) of high degrees have extremely small values near the endpoints (we call this property “rapid decay near the endpoints”) but extremely large values between these grid points and their roots are very close to the grid points near the endpoints. These results imply important general boundary effects for stable linear polynomial filters (we call this property “rapid boundary attenuation”).</div><div>Our results give interesting examples of nontrivial asymptotics of practically important solutions of special second-order linear recurrencies with polynomial coefficients studied by M.Petkovšek; to his memory we dedicate this paper.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102386"},"PeriodicalIF":0.6,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142422635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-26DOI: 10.1016/j.jsc.2024.102389
Éric Schost , Catherine St-Pierre
Let be a domain, with a maximal ideal, and let be any finite generating set of an ideal with finitely many roots (in an algebraic closure of the fraction field of ). We present a randomized -adic algorithm to recover the lexicographic Gröbner basis of , or of its primary component at the origin. We observe that previous results of Lazard's that use Hermite normal forms to compute Gröbner bases for ideals with two generators can be generalized to a generating set of cardinality greater than two. We use this result to bound the size of the coefficients of , and to control the probability of choosing a good maximal ideal . We give a complete cost analysis over number fields () and function fields (), and we obtain a complexity that is less than cubic in terms of the dimension of and softly linear in the size of its coefficients.
设 A 是一个域，m⊆A 是一个最大理想，设 F⊆A[x,y]是具有有限多个根（在 A 的分数域 K 的代数闭包中）的理想的任意有限生成集。我们提出了一种随机 m-adic 算法来恢复〈F〉⊆K[x,y]的词典格罗伯纳基 G 或其在原点的主成分。我们注意到，拉扎德之前利用赫米特正则表达式计算有两个生成子的理想的格罗伯纳基的结果，可以推广到心数大于两个的生成集 F。我们利用这一结果来约束 G 的系数大小，并控制选择一个好的最大理想 m⊆A 的概率。我们对数域（K=Q(α)）和函数域（）进行了完整的代价分析，得到的复杂度小于 K/〈G〉维数的立方，与其系数的大小呈软线性关系。
{"title":"An m-adic algorithm for bivariate Gröbner bases","authors":"Éric Schost , Catherine St-Pierre","doi":"10.1016/j.jsc.2024.102389","DOIUrl":"10.1016/j.jsc.2024.102389","url":null,"abstract":"<div><div>Let <span><math><mi>A</mi></math></span> be a domain, with <span><math><mi>m</mi><mo>⊆</mo><mi>A</mi></math></span> a maximal ideal, and let <span><math><mi>F</mi><mo>⊆</mo><mi>A</mi><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo></math></span> be any finite generating set of an ideal with finitely many roots (in an algebraic closure of the fraction field <span><math><mi>K</mi></math></span> of <span><math><mi>A</mi></math></span>). We present a randomized <span><math><mi>m</mi></math></span>-adic algorithm to recover the lexicographic Gröbner basis <span><math><mi>G</mi></math></span> of <span><math><mo>〈</mo><mi>F</mi><mo>〉</mo><mo>⊆</mo><mi>K</mi><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo></math></span>, or of its primary component at the origin. We observe that previous results of Lazard's that use Hermite normal forms to compute Gröbner bases for ideals with two generators can be generalized to a generating set <span><math><mi>F</mi></math></span> of cardinality greater than two. We use this result to bound the size of the coefficients of <span><math><mi>G</mi></math></span>, and to control the probability of choosing a <em>good</em> maximal ideal <span><math><mi>m</mi><mo>⊆</mo><mi>A</mi></math></span>. We give a complete cost analysis over number fields (<span><math><mi>K</mi><mo>=</mo><mi>Q</mi><mo>(</mo><mi>α</mi><mo>)</mo></math></span>) and function fields (<figure><img></figure>), and we obtain a complexity that is less than cubic in terms of the dimension of <span><math><mi>K</mi><mo>/</mo><mo>〈</mo><mi>G</mi><mo>〉</mo></math></span> and softly linear in the size of its coefficients.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102389"},"PeriodicalIF":0.6,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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