Several algorithms in computer algebra involve the computation of a power series solution of a given ordinary differential equation. Over finite fields, the problem is often lifted in an approximate $p$-adic setting to be well-posed. This raises precision concerns: how much precision do we need on the input to compute the output accurately? In the case of ordinary differential equations with separation of variables, we make use of the recent technique of differential precision to obtain optimal bounds on the stability of the Newton iteration. The results apply, for example, to algorithms for manipulating algebraic numbers over finite fields, for computing isogenies between elliptic curves or for deterministically finding roots of polynomials in finite fields. The new bounds lead to significant speedups in practice.
{"title":"On p-Adic Differential Equations with Separation of Variables","authors":"Pierre Lairez, Tristan Vaccon","doi":"10.1145/2930889.2930912","DOIUrl":"https://doi.org/10.1145/2930889.2930912","url":null,"abstract":"Several algorithms in computer algebra involve the computation of a power series solution of a given ordinary differential equation. Over finite fields, the problem is often lifted in an approximate $p$-adic setting to be well-posed. This raises precision concerns: how much precision do we need on the input to compute the output accurately? In the case of ordinary differential equations with separation of variables, we make use of the recent technique of differential precision to obtain optimal bounds on the stability of the Newton iteration. The results apply, for example, to algorithms for manipulating algebraic numbers over finite fields, for computing isogenies between elliptic curves or for deterministically finding roots of polynomials in finite fields. The new bounds lead to significant speedups in practice.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133922965","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 compute an upper bound for the orders of derivatives in the Rosenfeld-Grobner algorithm. This algorithm computes a regular decomposition of a radical differential ideal in the ring of differential polynomials over a differential field of characteristic zero with an arbitrary number of commuting derivations. This decomposition can then be used to test for membership in the given radical differential ideal. In particular, this algorithm allows us to determine whether a system of polynomial PDEs is consistent. Previously, the only known order upper bound was given by Golubitsky, Kondratieva, Moreno Maza, and Ovchinnikov for the case of a single derivation. We achieve our bound by associating to the algorithm antichain sequences whose lengths can be bounded using the results of Leon Sanchez and Ovchinnikov.
{"title":"Bounds for Orders of Derivatives in Differential Elimination Algorithms","authors":"Richard Gustavson, A. Ovchinnikov, G. Pogudin","doi":"10.1145/2930889.2930922","DOIUrl":"https://doi.org/10.1145/2930889.2930922","url":null,"abstract":"We compute an upper bound for the orders of derivatives in the Rosenfeld-Grobner algorithm. This algorithm computes a regular decomposition of a radical differential ideal in the ring of differential polynomials over a differential field of characteristic zero with an arbitrary number of commuting derivations. This decomposition can then be used to test for membership in the given radical differential ideal. In particular, this algorithm allows us to determine whether a system of polynomial PDEs is consistent. Previously, the only known order upper bound was given by Golubitsky, Kondratieva, Moreno Maza, and Ovchinnikov for the case of a single derivation. We achieve our bound by associating to the algorithm antichain sequences whose lengths can be bounded using the results of Leon Sanchez and Ovchinnikov.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129919513","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 solve the existence problem of telescopers for rational functions in three discrete variables. We reduce the problem to that of deciding the summability of bivariate rational functions, a problem which has recently been solved. This existence criteria is used, for example, for detecting the termination of Zeilberger's algorithm to the function classes studied in this paper.
{"title":"Existence Problem of Telescopers: Beyond the Bivariate Case","authors":"Shaoshi Chen, Q. Hou, G. Labahn, Rong-Hua Wang","doi":"10.1145/2930889.2930895","DOIUrl":"https://doi.org/10.1145/2930889.2930895","url":null,"abstract":"In this paper, we solve the existence problem of telescopers for rational functions in three discrete variables. We reduce the problem to that of deciding the summability of bivariate rational functions, a problem which has recently been solved. This existence criteria is used, for example, for detecting the termination of Zeilberger's algorithm to the function classes studied in this paper.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115921851","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 study a variant of the Miller-Rabin primality test, which only looks at the last (z+1) powers of the base. This test is between Miller-Rabin and Fermat in terms of strength. For (z=1) the test can be thought of as a variant of the Solovay-Strassen test. We show that for every (z ≥ 0) this test has infinitely many "Carmichael" numbers. We also give empirical results on the rate of growth of the test's "Carmichael" numbers, noting that the growth rate decreases geometrically with increasing (z). We provide some heuristic evidence for this pattern. We also extend our existence result to some generalizations of Miller-Rabin that use (b)-th powers instead of squares.
{"title":"Infinitely Many Carmichael Numbers for a Modified Miller-Rabin Prime Test","authors":"E. Bach, R. Fernando","doi":"10.1145/2930889.2930911","DOIUrl":"https://doi.org/10.1145/2930889.2930911","url":null,"abstract":"We study a variant of the Miller-Rabin primality test, which only looks at the last (z+1) powers of the base. This test is between Miller-Rabin and Fermat in terms of strength. For (z=1) the test can be thought of as a variant of the Solovay-Strassen test. We show that for every (z ≥ 0) this test has infinitely many \"Carmichael\" numbers. We also give empirical results on the rate of growth of the test's \"Carmichael\" numbers, noting that the growth rate decreases geometrically with increasing (z). We provide some heuristic evidence for this pattern. We also extend our existence result to some generalizations of Miller-Rabin that use (b)-th powers instead of squares.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125988200","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}
Ore operators form a common algebraic abstraction of linear ordinary differential and recurrence equations. Given an Ore operator L with polynomial coefficients in x, it generates a left ideal I in the Ore algebra over the field k(x) of rational functions. We present an algorithm for computing a basis of the contraction ideal of I in the Ore algebra over the ring R[x] of polynomials, where~$R$ may be either k or a domain with k as its fraction field. This algorithm is based on recent work on desingularization for Ore operators by Chen, Jaroschek, Kauers and Singer. Using a basis of the contraction ideal, we compute a completely desingularized operator for L whose leading coefficient not only has minimal degree in x but also has minimal content. Completely desingularized operators have interesting applications such as certifying integer sequences and checking special cases of a conjecture of Krattenthaler.
{"title":"Contraction of Ore Ideals with Applications","authors":"Yi Zhang","doi":"10.1145/2930889.2930890","DOIUrl":"https://doi.org/10.1145/2930889.2930890","url":null,"abstract":"Ore operators form a common algebraic abstraction of linear ordinary differential and recurrence equations. Given an Ore operator L with polynomial coefficients in x, it generates a left ideal I in the Ore algebra over the field k(x) of rational functions. We present an algorithm for computing a basis of the contraction ideal of I in the Ore algebra over the ring R[x] of polynomials, where~$R$ may be either k or a domain with k as its fraction field. This algorithm is based on recent work on desingularization for Ore operators by Chen, Jaroschek, Kauers and Singer. Using a basis of the contraction ideal, we compute a completely desingularized operator for L whose leading coefficient not only has minimal degree in x but also has minimal content. Completely desingularized operators have interesting applications such as certifying integer sequences and checking special cases of a conjecture of Krattenthaler.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133964989","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}
Similarly to the correspondence between radical ideals of a polynomial ring and varieties in algebraic geometry, a correspondence between radical differential ideals and their analytic solution sets has been established in differential algebra. This tutorial discusses aspects of this correspondence involving symbolic computation. In particular, an introduction to the Thomas decomposition method is given. It splits a system of polynomially nonlinear partial differential equations into finitely many so-called simple differential systems whose solution sets form a partition of the original solution set. The power series solutions of each simple system can be determined in a straightforward way. Conversely, certain sets of analytic functions admit an implicit description in terms of partial differential equations and inequations. Strategies for solving related differential elimination problems and applications to symbolic solving of differential equations are presented. A Maple implementation of the Thomas decomposition method is freely available.
{"title":"Formal Algorithmic Elimination for PDEs","authors":"D. Robertz","doi":"10.1145/2930889.2930941","DOIUrl":"https://doi.org/10.1145/2930889.2930941","url":null,"abstract":"Similarly to the correspondence between radical ideals of a polynomial ring and varieties in algebraic geometry, a correspondence between radical differential ideals and their analytic solution sets has been established in differential algebra. This tutorial discusses aspects of this correspondence involving symbolic computation. In particular, an introduction to the Thomas decomposition method is given. It splits a system of polynomially nonlinear partial differential equations into finitely many so-called simple differential systems whose solution sets form a partition of the original solution set. The power series solutions of each simple system can be determined in a straightforward way. Conversely, certain sets of analytic functions admit an implicit description in terms of partial differential equations and inequations. Strategies for solving related differential elimination problems and applications to symbolic solving of differential equations are presented. A Maple implementation of the Thomas decomposition method is freely available.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116171076","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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