Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials
Two problems are addressed: reduction of an arbitrary degree non-special divisor to the equivalent divisor of the degree equal to the genus of a curve, and addition of divisors of arbitrary degrees. The hyperelliptic case is considered the simplest model. Explicit formulas defining reduced divisors...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2020 |
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2020
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/210697 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials. Julia Bernatska and Yaacov Kopeliovich. SIGMA 16 (2020), 053, 21 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862738169492930560 |
|---|---|
| author | Bernatska, Julia Kopeliovich, Yaacov |
| author_facet | Bernatska, Julia Kopeliovich, Yaacov |
| citation_txt | Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials. Julia Bernatska and Yaacov Kopeliovich. SIGMA 16 (2020), 053, 21 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Two problems are addressed: reduction of an arbitrary degree non-special divisor to the equivalent divisor of the degree equal to the genus of a curve, and addition of divisors of arbitrary degrees. The hyperelliptic case is considered the simplest model. Explicit formulas defining reduced divisors for some particular cases are found. The reduced divisors are obtained in the form of a solution of the Jacobi inversion problem, which provides a way of computing Abelian functions on arbitrary non-special divisors. An effective reduction algorithm is proposed, which has the advantage that it involves only arithmetic operations on polynomials. The proposed addition algorithm contains more details compared with the known in cryptography, and is extended to divisors of arbitrary degrees compared with the known in the theory of hyperelliptic functions.
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| first_indexed | 2025-12-17T12:04:31Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-210697 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:31Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bernatska, Julia Kopeliovich, Yaacov 2025-12-15T15:23:06Z 2020 Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials. Julia Bernatska and Yaacov Kopeliovich. SIGMA 16 (2020), 053, 21 pages 1815-0659 2020 Mathematics Subject Classification: 32Q30; 14G50 arXiv:1912.13277 https://nasplib.isofts.kiev.ua/handle/123456789/210697 https://doi.org/10.3842/SIGMA.2020.053 Two problems are addressed: reduction of an arbitrary degree non-special divisor to the equivalent divisor of the degree equal to the genus of a curve, and addition of divisors of arbitrary degrees. The hyperelliptic case is considered the simplest model. Explicit formulas defining reduced divisors for some particular cases are found. The reduced divisors are obtained in the form of a solution of the Jacobi inversion problem, which provides a way of computing Abelian functions on arbitrary non-special divisors. An effective reduction algorithm is proposed, which has the advantage that it involves only arithmetic operations on polynomials. The proposed addition algorithm contains more details compared with the known in cryptography, and is extended to divisors of arbitrary degrees compared with the known in the theory of hyperelliptic functions. The authors are thankful to the referees for the comments that have improved the paper substantially. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials Article published earlier |
| spellingShingle | Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials Bernatska, Julia Kopeliovich, Yaacov |
| title | Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials |
| title_full | Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials |
| title_fullStr | Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials |
| title_full_unstemmed | Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials |
| title_short | Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials |
| title_sort | addition of divisors on hyperelliptic curves via interpolation polynomials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210697 |
| work_keys_str_mv | AT bernatskajulia additionofdivisorsonhyperellipticcurvesviainterpolationpolynomials AT kopeliovichyaacov additionofdivisorsonhyperellipticcurvesviainterpolationpolynomials |