Dancing samba with Ramanujan partition congruences
The article presents an algorithm to compute a C[t]-module basis G for a given subalgebra A over a polynomial ring R=C[x] with a Euclidean domain C as the domain of coefficients and t a given element of A. The reduction modulo G allows a subalgebra membership test. The algorithm also works for more...
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| Published in | Journal of symbolic computation Vol. 84; pp. 14 - 24 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Ltd
01.01.2018
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| Subjects | |
| Online Access | Get full text |
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| Abstract | The article presents an algorithm to compute a C[t]-module basis G for a given subalgebra A over a polynomial ring R=C[x] with a Euclidean domain C as the domain of coefficients and t a given element of A. The reduction modulo G allows a subalgebra membership test. The algorithm also works for more general rings R, in particular for a ring R⊂C((q)) with the property that f∈R is zero if and only if the order of f is positive. As an application, we algorithmically derive an explicit identity (in terms of quotients of Dedekind η-functions and Klein's j-invariant) that shows that p(11n+6) is divisible by 11 for every natural number n where p(n) denotes the number of partitions of n. |
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| AbstractList | The article presents an algorithm to compute a C[t]-module basis G for a given subalgebra A over a polynomial ring R=C[x] with a Euclidean domain C as the domain of coefficients and t a given element of A. The reduction modulo G allows a subalgebra membership test. The algorithm also works for more general rings R, in particular for a ring R⊂C((q)) with the property that f∈R is zero if and only if the order of f is positive. As an application, we algorithmically derive an explicit identity (in terms of quotients of Dedekind η-functions and Klein's j-invariant) that shows that p(11n+6) is divisible by 11 for every natural number n where p(n) denotes the number of partitions of n. |
| Author | Hemmecke, Ralf |
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| Cites_doi | 10.1016/j.jsc.2014.09.018 10.1007/BF01378341 10.1007/978-1-4684-9884-4 10.2307/2371972 |
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| Keywords | Number theoretic algorithm Partition identities Subalgebra basis |
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| References | Ramanujan (br0070) 1919; 19 Lehner (br0030) 1943; 65 Radu (br0060) 2015; 68 Ramanujan (br0080) 1921; 9 Paule, Radu (br0100) 2017 Serre (br0090) 1973 br0020 Paule, Radu (br0040) 2015 Becker, Weispfenning (br0010) 1993; vol. 141 Lehner (10.1016/j.jsc.2017.02.001_br0030) 1943; 65 Ramanujan (10.1016/j.jsc.2017.02.001_br0080) 1921; 9 Serre (10.1016/j.jsc.2017.02.001_br0090) 1973 Paule (10.1016/j.jsc.2017.02.001_br0040) 2015 Becker (10.1016/j.jsc.2017.02.001_br0010) 1993; vol. 141 Ramanujan (10.1016/j.jsc.2017.02.001_br0070) 1919; 19 Paule (10.1016/j.jsc.2017.02.001_br0100) 2017 Radu (10.1016/j.jsc.2017.02.001_br0060) 2015; 68 |
| References_xml | – volume: vol. 141 year: 1993 ident: br0010 article-title: Gröbner Bases. A Computational Approach to Commutative Algebra publication-title: Graduate Texts in Mathematics contributor: fullname: Weispfenning – year: 2017 ident: br0100 article-title: A new witness identity for publication-title: Number Theory in Honor of Krishna Alladi's 60th Birthday contributor: fullname: Radu – volume: 19 start-page: 207 year: 1919 end-page: 210 ident: br0070 article-title: Some properties of publication-title: Proc. Camb. Philos. Soc. contributor: fullname: Ramanujan – ident: br0020 article-title: FriCAS – an advanced computer algebra system – volume: 68 start-page: 225 year: 2015 end-page: 253 ident: br0060 article-title: An algorithmic approach to Ramanujan–Kolberg identities publication-title: J. Symb. Comput. contributor: fullname: Radu – volume: 9 start-page: 147 year: 1921 end-page: 153 ident: br0080 article-title: Congruence properties of partitions publication-title: Math. Z. contributor: fullname: Ramanujan – volume: 65 start-page: 492 year: 1943 end-page: 520 ident: br0030 article-title: Ramanujan identities involving the partition function for the moduli publication-title: Am. J. Math. contributor: fullname: Lehner – year: 1973 ident: br0090 article-title: A Course in Arithmetic publication-title: Graduate Texts in Mathematics contributor: fullname: Serre – start-page: 511 year: 2015 end-page: 544 ident: br0040 article-title: Partition analysis, modular functions, and computer algebra publication-title: Recent Trends in Combinatorics contributor: fullname: Radu – volume: 68 start-page: 225 issue: 1 year: 2015 ident: 10.1016/j.jsc.2017.02.001_br0060 article-title: An algorithmic approach to Ramanujan–Kolberg identities publication-title: J. Symb. Comput. doi: 10.1016/j.jsc.2014.09.018 contributor: fullname: Radu – volume: 9 start-page: 147 issue: 1–2 year: 1921 ident: 10.1016/j.jsc.2017.02.001_br0080 article-title: Congruence properties of partitions publication-title: Math. Z. doi: 10.1007/BF01378341 contributor: fullname: Ramanujan – year: 1973 ident: 10.1016/j.jsc.2017.02.001_br0090 article-title: A Course in Arithmetic doi: 10.1007/978-1-4684-9884-4 contributor: fullname: Serre – volume: 65 start-page: 492 issue: 3 year: 1943 ident: 10.1016/j.jsc.2017.02.001_br0030 article-title: Ramanujan identities involving the partition function for the moduli 11a publication-title: Am. J. Math. doi: 10.2307/2371972 contributor: fullname: Lehner – volume: vol. 141 year: 1993 ident: 10.1016/j.jsc.2017.02.001_br0010 article-title: Gröbner Bases. A Computational Approach to Commutative Algebra contributor: fullname: Becker – start-page: 511 year: 2015 ident: 10.1016/j.jsc.2017.02.001_br0040 article-title: Partition analysis, modular functions, and computer algebra contributor: fullname: Paule – volume: 19 start-page: 207 year: 1919 ident: 10.1016/j.jsc.2017.02.001_br0070 article-title: Some properties of p(n), the number of partitions of n publication-title: Proc. Camb. Philos. Soc. contributor: fullname: Ramanujan – year: 2017 ident: 10.1016/j.jsc.2017.02.001_br0100 article-title: A new witness identity for 11|p(11n+6) contributor: fullname: Paule |
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| SubjectTerms | Number theoretic algorithm Partition identities Subalgebra basis |
| Title | Dancing samba with Ramanujan partition congruences |
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