Pub Date : 2024-10-21DOI: 10.1016/j.jaca.2024.100026
Andrea Lesavourey , Thomas Plantard , Willy Susilo
We explore a fairly generic method to compute roots of polynomials over number fields through complex embeddings. Our main contribution is to show how to use a structure of a relative extension to decode in a subfield. Additionally we describe several heuristic options to improve practical efficiency. We provide experimental data from our implementation and compare our methods to the state of the art algorithm implemented in Pari/Gp.
{"title":"Improved computation of polynomial roots over number fields when using complex embeddings","authors":"Andrea Lesavourey , Thomas Plantard , Willy Susilo","doi":"10.1016/j.jaca.2024.100026","DOIUrl":"10.1016/j.jaca.2024.100026","url":null,"abstract":"<div><div>We explore a fairly generic method to compute roots of polynomials over number fields through complex embeddings. Our main contribution is to show how to use a structure of a relative extension to decode in a subfield. Additionally we describe several heuristic options to improve practical efficiency. We provide experimental data from our implementation and compare our methods to the state of the art algorithm implemented in <span>Pari/Gp</span>.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"12 ","pages":"Article 100026"},"PeriodicalIF":0.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142538675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-21DOI: 10.1016/j.jaca.2024.100027
Masayuki Noro
The notion of the compatibility between a term order in a polynomial ring R and a module term order in is crucial to ensure the termination of a signature-based algorithm for general input ideals. However, it is shown experimentally that the compatibility does not necessarily imply efficient computation. Our experiments show that combining non-compatible term orders can improve performance for computing Gröbner bases with respect to some term orders. In such cases, we can use the Hilbert function to guarantee the termination. The Hilbert function can be computed by using a Gröbner basis with respect to some term order and thus the resulting algorithm is considered a change of ordering algorithm. In this paper, we give the details of the new change of ordering algorithm and we compare its performance with that of the usual Hilbert-driven Buchberger algorithm and the Gröbner walk algorithm.
多项式环 R 中的阶次与 Rl 中的模块阶次之间的兼容性概念对于确保基于签名的一般输入理想算法的终止至关重要。然而，实验表明，兼容性并不一定意味着高效计算。我们的实验表明，结合不兼容的项阶可以提高计算格罗伯纳基时某些项阶的性能。在这种情况下，我们可以使用希尔伯特函数来保证终止。希尔伯特函数可以通过使用相对于某些术语阶的格罗伯纳基计算出来，因此由此产生的算法被认为是一种改变阶算法。在本文中，我们给出了新的改序算法的细节，并将其性能与通常的希尔伯特驱动布赫伯格算法和格罗伯纳行走算法进行了比较。
{"title":"Signature-based algorithm under non-compatible term orders and its application to change of ordering","authors":"Masayuki Noro","doi":"10.1016/j.jaca.2024.100027","DOIUrl":"10.1016/j.jaca.2024.100027","url":null,"abstract":"<div><div>The notion of the compatibility between a term order in a polynomial ring <em>R</em> and a module term order in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>l</mi></mrow></msup></math></span> is crucial to ensure the termination of a signature-based algorithm for general input ideals. However, it is shown experimentally that the compatibility does not necessarily imply efficient computation. Our experiments show that combining non-compatible term orders can improve performance for computing Gröbner bases with respect to some term orders. In such cases, we can use the Hilbert function to guarantee the termination. The Hilbert function can be computed by using a Gröbner basis with respect to some term order and thus the resulting algorithm is considered a change of ordering algorithm. In this paper, we give the details of the new change of ordering algorithm and we compare its performance with that of the usual Hilbert-driven Buchberger algorithm and the Gröbner walk algorithm.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"12 ","pages":"Article 100027"},"PeriodicalIF":0.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-21DOI: 10.1016/j.jaca.2024.100025
Plamen Simeonov , Ron Goldman
We introduce and study the properties of new negative degree rational Bernstein bases associated with the Askey–Wilson operator and we use these bases to define new types of rational Bernstein-Bézier curves. We also introduce a new type of blossom, the multirational Askey–Wilson blossom. We prove that four axioms uniquely characterize this blossom and we provide an explicit formula for this multirational blossom involving a right inverse of the Askey–Wilson operator. A formula for the coefficients of a function expanded in a rational Askey–Wilson Bernstein basis in terms of certain values of the Askey–Wilson operator is derived. We also establish a dual functional property that expresses the coefficients of these new types of rational Bernstein–Bézier curves in terms of values of their multirational Askey–Wilson blossom.
{"title":"Rational Askey–Wilson Bernstein bases and a multirational Askey–Wilson blossom","authors":"Plamen Simeonov , Ron Goldman","doi":"10.1016/j.jaca.2024.100025","DOIUrl":"10.1016/j.jaca.2024.100025","url":null,"abstract":"<div><div>We introduce and study the properties of new negative degree rational Bernstein bases associated with the Askey–Wilson operator and we use these bases to define new types of rational Bernstein-Bézier curves. We also introduce a new type of blossom, the <em>multirational Askey–Wilson blossom</em>. We prove that four axioms uniquely characterize this blossom and we provide an explicit formula for this multirational blossom involving a right inverse of the Askey–Wilson operator. A formula for the coefficients of a function expanded in a rational Askey–Wilson Bernstein basis in terms of certain values of the Askey–Wilson operator is derived. We also establish a dual functional property that expresses the coefficients of these new types of rational Bernstein–Bézier curves in terms of values of their multirational Askey–Wilson blossom.</div></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"12 ","pages":"Article 100025"},"PeriodicalIF":0.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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