{"title":"Numerical analysis for electromagnetic scattering from nonlinear boundary conditions","authors":"Jörg Nick","doi":"10.1090/mcom/3914","DOIUrl":"https://doi.org/10.1090/mcom/3914","url":null,"abstract":"","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135579674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A kit for linear forms in three logarithms","authors":"Maurice Mignotte, Paul Voutier, Michel Laurent","doi":"10.1090/mcom/3908","DOIUrl":"https://doi.org/10.1090/mcom/3908","url":null,"abstract":"","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135096432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
V. Arul, J. Booher, Steven R. Groen, Everett W. Howe, Wanlin Li, Vlad Matei, R. Pries, Caleb Springer
We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose CC and C′C’ are curves over a finite field KK, with KK-rational base points PP and P′P’, and let DD and D′D’ be the pullbacks (via the Abel–Jacobi map) of the multiplication-by-22 map
我们研究了有限域上的曲线在多大程度上由它们的ζ函数和它们的某些覆盖的ζ功能表征。假设C C和C′C′是有限域K K上的曲线，具有K K-有理基点P P和P′P′，并且设D D和D′D′是乘-2 2映射在其雅可比上的回调（通过Abel–Jacobi映射）。如果J a C（C）Jac（C）和J a CK与J（D）Jac（D）和J（D′）Jac。对于自同构群包含八阶二面体群的亏格2 2的曲线，我们证明了双同构曲线对的数量大于天真启发式预测的数量，并对这一现象给出了解释。
{"title":"Doubly isogenous genus-2 curves with 𝐷₄-action","authors":"V. Arul, J. Booher, Steven R. Groen, Everett W. Howe, Wanlin Li, Vlad Matei, R. Pries, Caleb Springer","doi":"10.1090/mcom/3891","DOIUrl":"https://doi.org/10.1090/mcom/3891","url":null,"abstract":"<p>We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C\">\u0000 <mml:semantics>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">C</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C prime\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mo>′</mml:mo>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C’</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> are curves over a finite field <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, with <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-rational base points <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P\">\u0000 <mml:semantics>\u0000 <mml:mi>P</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">P</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P prime\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>P</mml:mi>\u0000 <mml:mo>′</mml:mo>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">P’</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, and let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\">\u0000 <mml:semantics>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">D</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D prime\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mo>′</mml:mo>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">D’</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be the pullbacks (via the Abel–Jacobi map) of the multiplication-by-<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semantics>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> map","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"1 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44511508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Building on Mazur’s 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q left-parenthesis StartRoot d EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msqrt> <mml:mi>d</mml:mi> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">mathbb {Q}(sqrt {d})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue d EndAbsoluteValue greater-than 10 Superscript 4"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>></mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">|d| > 10^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="19"> <mml:semantics> <mml:mn>19</mml:mn> <mml:annotation encoding="application/x-tex">19</mml:annotation> </mml:semantics> </mml:math> </inline-formula> quadratic fields, including <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q left-parenthesis StartRoot 213 EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msqrt> <mml:mn>213</mml:mn> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">mathbb {Q}(sqrt {213})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a
{"title":"Cyclic isogenies of elliptic curves over fixed quadratic fields","authors":"Barinder Banwait, Filip Najman, Oana Padurariu","doi":"10.1090/mcom/3894","DOIUrl":"https://doi.org/10.1090/mcom/3894","url":null,"abstract":"Building on Mazur’s 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q left-parenthesis StartRoot d EndRoot right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msqrt> <mml:mi>d</mml:mi> </mml:msqrt> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathbb {Q}(sqrt {d})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue d EndAbsoluteValue greater-than 10 Superscript 4\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mo>></mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">|d| > 10^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"19\"> <mml:semantics> <mml:mn>19</mml:mn> <mml:annotation encoding=\"application/x-tex\">19</mml:annotation> </mml:semantics> </mml:math> </inline-formula> quadratic fields, including <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q left-parenthesis StartRoot 213 EndRoot right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msqrt> <mml:mn>213</mml:mn> </mml:msqrt> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathbb {Q}(sqrt {213})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136037366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In solid state physics, electronic properties of crystalline materials are often inferred from the spectrum of periodic Schrödinger operators. As a consequence of Bloch’s theorem, the numerical computation of electronic quantities of interest involves computing derivatives or integrals over the Brillouin zone of so-called energy bands, which are piecewise smooth, Lipschitz continuous periodic functions obtained by solving a parametrized elliptic eigenvalue problem on a Hilbert space of periodic functions. Classical discretization strategies for resolving these eigenvalue problems produce approximate energy bands that are either non-periodic or discontinuous, both of which cause difficulty when computing numerical derivatives or employing numerical quadrature. In this article, we study an alternative discretization strategy based on an ad hoc operator modification approach. While specific instances of this approach have been proposed in the physics literature, we introduce here a systematic formulation of this operator modification approach. We derive a priori error estimates for the resulting energy bands and we show that these bands are periodic and can be made arbitrarily smooth (away from band crossings) by adjusting suitable parameters in the operator modification approach. Numerical experiments involving a toy model in 1D, graphene in 2D, and silicon in 3D validate our theoretical results and showcase the efficiency of the operator modification approach.
{"title":"Modified-operator method for the calculation of band diagrams of crystalline materials","authors":"Eric Cancès, Muhammad Hassan, Laurent Vidal","doi":"10.1090/mcom/3897","DOIUrl":"https://doi.org/10.1090/mcom/3897","url":null,"abstract":"In solid state physics, electronic properties of crystalline materials are often inferred from the spectrum of periodic Schrödinger operators. As a consequence of Bloch’s theorem, the numerical computation of electronic quantities of interest involves computing derivatives or integrals over the Brillouin zone of so-called energy bands, which are piecewise smooth, Lipschitz continuous periodic functions obtained by solving a parametrized elliptic eigenvalue problem on a Hilbert space of periodic functions. Classical discretization strategies for resolving these eigenvalue problems produce approximate energy bands that are either non-periodic or discontinuous, both of which cause difficulty when computing numerical derivatives or employing numerical quadrature. In this article, we study an alternative discretization strategy based on an ad hoc operator modification approach. While specific instances of this approach have been proposed in the physics literature, we introduce here a systematic formulation of this operator modification approach. We derive a priori error estimates for the resulting energy bands and we show that these bands are periodic and can be made arbitrarily smooth (away from band crossings) by adjusting suitable parameters in the operator modification approach. Numerical experiments involving a toy model in 1D, graphene in 2D, and silicon in 3D validate our theoretical results and showcase the efficiency of the operator modification approach.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"184 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135236031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a revisit of the seeds algorithm to explore the semigroup tree. First, an equivalent definition of seed is presented, which seems easier to manage. Second, we determine the seeds of semigroups with at most three left elements. And third, we find the great-grandchildren of any numerical semigroup in terms of its seeds. The the right-generators descendant (RGD) algorithm is the fastest known algorithm at the moment. But if one compares the originary seeds algorithm with the RGD algorithm, one observes that the seeds algorithm uses more elaborated mathematical tools while the RGD algorithm uses data structures that are better adapted to the final C implementations. For genera up to around one half of the maximum size of native integers, the newly defined seeds algorithm performs significantly better than the RGD algorithm. For future compilators allowing larger native sized integers this may constitute a powerful tool to explore the semigroup tree up to genera never explored before. The new seeds algorithm uses bitwise integer operations, the knowledge of the seeds of semigroups with at most three left elements and of the great-grandchildren of any numerical semigroup, apart from techniques such as parallelization and depth first search as wisely introduced in this context by Fromentin and Hivert [Math. Comp. 85 (2016) pp. 2553–2568]. The algorithm has been used to prove that there are no Eliahou semigroups of genus 6666, hence proving the Wilf conjecture for genus up to 6666. We also found three Eliahou semigroups of genus 6767. One of these semigroups is neither of Eliahou-Fromentin type, nor of Delgado’s type. However, it is a member of a new family suggested by Shalom Eliahou.
我们提出了一种种子算法来探索半群树。首先，给出了种子的等价定义，该定义似乎更易于管理。其次，我们确定了最多有三个左元素的半群的种子。第三，我们根据它的种子找到任何数值半群的曾孙。右生成器后代(RGD)算法是目前已知最快的算法。但是，如果将原始种子算法与RGD算法进行比较，就会发现种子算法使用了更复杂的数学工具，而RGD算法使用了更适合最终C实现的数据结构。对于原生整数最大大小的一半左右的属，新定义的种子算法的性能明显优于RGD算法。对于将来允许更大的本地整数的编译器来说，这可能是一个强大的工具，可以探索半群树，直到以前从未探索过的属。新的种子算法使用位整数运算，除了并行化和深度优先搜索等技术(由Fromentin和Hivert [Math]在此背景下明智地引入)之外，还使用最多有三个左元素的半群的种子和任何数值半群的曾孙的知识。Comp. 85 (2016) pp. 2553-2568]。利用该算法证明了66 66属的Eliahou半群不存在，从而证明了66 66属的Wilf猜想。我们还发现了3个属667的Eliahou半群。其中一个半群既不是Eliahou-Fromentin型，也不是Delgado型。然而，它是Shalom Eliahou建议的一个新家庭的成员。
{"title":"On the seeds and the great-grandchildren of a numerical semigroup","authors":"Maria Bras-Amorós","doi":"10.1090/mcom/3881","DOIUrl":"https://doi.org/10.1090/mcom/3881","url":null,"abstract":"We present a revisit of the seeds algorithm to explore the semigroup tree. First, an equivalent definition of seed is presented, which seems easier to manage. Second, we determine the seeds of semigroups with at most three left elements. And third, we find the great-grandchildren of any numerical semigroup in terms of its seeds. The the right-generators descendant (RGD) algorithm is the fastest known algorithm at the moment. But if one compares the originary seeds algorithm with the RGD algorithm, one observes that the seeds algorithm uses more elaborated mathematical tools while the RGD algorithm uses data structures that are better adapted to the final C implementations. For genera up to around one half of the maximum size of native integers, the newly defined seeds algorithm performs significantly better than the RGD algorithm. For future compilators allowing larger native sized integers this may constitute a powerful tool to explore the semigroup tree up to genera never explored before. The new seeds algorithm uses bitwise integer operations, the knowledge of the seeds of semigroups with at most three left elements and of the great-grandchildren of any numerical semigroup, apart from techniques such as parallelization and depth first search as wisely introduced in this context by Fromentin and Hivert [Math. Comp. 85 (2016) pp. 2553–2568]. The algorithm has been used to prove that there are no Eliahou semigroups of genus <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"66\"> <mml:semantics> <mml:mn>66</mml:mn> <mml:annotation encoding=\"application/x-tex\">66</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, hence proving the Wilf conjecture for genus up to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"66\"> <mml:semantics> <mml:mn>66</mml:mn> <mml:annotation encoding=\"application/x-tex\">66</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also found three Eliahou semigroups of genus <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"67\"> <mml:semantics> <mml:mn>67</mml:mn> <mml:annotation encoding=\"application/x-tex\">67</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. One of these semigroups is neither of Eliahou-Fromentin type, nor of Delgado’s type. However, it is a member of a new family suggested by Shalom Eliahou.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136337533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Over the last several decades, improvements in the fields of analytic combinatorics and computer algebra have made determining the asymptotic behaviour of sequences satisfying linear recurrence relations with polynomial coefficients largely a matter of routine, under assumptions that hold often in practice. The algorithms involved typically take a sequence, encoded by a recurrence relation and initial terms, and return the leading terms in an asymptotic expansion up to a big-O error term. Less studied, however, are effective techniques giving an explicit bound on asymptotic error terms. Among other things, such explicit bounds typically allow the user to automatically prove sequence positivity (an active area of enumerative and algebraic combinatorics) by exhibiting an index when positive leading asymptotic behaviour dominates any error terms. In this article, we present a practical algorithm for computing such asymptotic approximations with rigorous error bounds, under the assumption that the generating series of the sequence is a solution of a differential equation with regular (Fuchsian) dominant singularities. Our algorithm approximately follows the singularity analysis method of Flajolet and Odlyzko, except that all big-O terms involved in the derivation of the asymptotic expansion are replaced by explicit error terms. The computation of the error terms combines analytic bounds from the literature with effective techniques from rigorous numerics and computer algebra. We implement our algorithm in the SageMath computer algebra system and exhibit its use on a variety of applications (including our original motivating example, solution uniqueness in the Canham model for the shape of genus one biomembranes).
{"title":"Computing error bounds for asymptotic expansions of regular P-recursive sequences","authors":"Ruiwen Dong, Stephen Melczer, Marc Mezzarobba","doi":"10.1090/mcom/3888","DOIUrl":"https://doi.org/10.1090/mcom/3888","url":null,"abstract":"Over the last several decades, improvements in the fields of analytic combinatorics and computer algebra have made determining the asymptotic behaviour of sequences satisfying linear recurrence relations with polynomial coefficients largely a matter of routine, under assumptions that hold often in practice. The algorithms involved typically take a sequence, encoded by a recurrence relation and initial terms, and return the leading terms in an asymptotic expansion up to a big-O error term. Less studied, however, are effective techniques giving an explicit bound on asymptotic error terms. Among other things, such explicit bounds typically allow the user to automatically prove sequence positivity (an active area of enumerative and algebraic combinatorics) by exhibiting an index when positive leading asymptotic behaviour dominates any error terms. In this article, we present a practical algorithm for computing such asymptotic approximations with rigorous error bounds, under the assumption that the generating series of the sequence is a solution of a differential equation with regular (Fuchsian) dominant singularities. Our algorithm approximately follows the singularity analysis method of Flajolet and Odlyzko, except that all big-O terms involved in the derivation of the asymptotic expansion are replaced by explicit error terms. The computation of the error terms combines analytic bounds from the literature with effective techniques from rigorous numerics and computer algebra. We implement our algorithm in the SageMath computer algebra system and exhibit its use on a variety of applications (including our original motivating example, solution uniqueness in the Canham model for the shape of genus one biomembranes).","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136337990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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