Computing Puiseux series: a fast divide and conquer algorithm
Let F ∈ [ X , Y ] be a polynomial of total degree D defined over a perfect field of characteristic zero or greater than D . Assuming F separable with respect to Y , we provide an algorithm that computes all singular parts of Puiseux series of F above X = 0 in an expected Ø ˜ ( D δ ) operations in ,...
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| Vydáno v: | Annales Henri Lebesgue Ročník 4; s. 1061 - 1102 |
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UFR de Mathématiques - IRMAR
2021
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| ISSN: | 2644-9463, 2644-9463 |
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| Abstract | Let F ∈ [ X , Y ] be a polynomial of total degree D defined over a perfect field of characteristic zero or greater than D . Assuming F separable with respect to Y , we provide an algorithm that computes all singular parts of Puiseux series of F above X = 0 in an expected Ø ˜ ( D δ ) operations in , where δ is the valuation of the resultant of F and its partial derivative with respect to Y . To this aim, we use a divide and conquer strategy and replace univariate factorisation by dynamic evaluation. As a first main corollary, we compute the irreducible factors of F in [ [ X ] ] [ Y ] up to an arbitrary precision X N with Ø ˜ ( D ( δ + N ) ) arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by F with Ø ˜ ( D 3 ) arithmetic operations and, if = ℚ , with Ø ˜ ( ( h + 1 ) D 3 ) bit operations using probabilistic algorithms, where h is the logarithmic height of F . |
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| AbstractList | Let $F ∈ K[X, Y ]$ be a polynomial of total degree D defined over a field K of characteristic zero or greater than D. Assuming F separable with respect to Y , we provide an algorithm that computes all Puiseux series of F above X = 0 in less than $O˜(D δ)$ operations in K, where δ is the valuation of the resultant of F and its partial derivative with respect to Y. To this aim, we use a divide and conquer strategy and replace univariate factorisation by dynamic evaluation. As a first main corollary, we compute the irreducible factors of F in $K[[X]][Y ]$ up to an arbitrary precision X N with $O˜(D(δ + N))$ arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by F with $O˜(D^3)$ arithmetic operations and, if K = Q, with $O˜((h+1) D^3)$ bit operations using probabilistic algorithms, where h is the logarithmic height of F . Let F ∈ [ X , Y ] be a polynomial of total degree D defined over a perfect field of characteristic zero or greater than D . Assuming F separable with respect to Y , we provide an algorithm that computes all singular parts of Puiseux series of F above X = 0 in an expected Ø ˜ ( D δ ) operations in , where δ is the valuation of the resultant of F and its partial derivative with respect to Y . To this aim, we use a divide and conquer strategy and replace univariate factorisation by dynamic evaluation. As a first main corollary, we compute the irreducible factors of F in [ [ X ] ] [ Y ] up to an arbitrary precision X N with Ø ˜ ( D ( δ + N ) ) arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by F with Ø ˜ ( D 3 ) arithmetic operations and, if = ℚ , with Ø ˜ ( ( h + 1 ) D 3 ) bit operations using probabilistic algorithms, where h is the logarithmic height of F . |
| Author | Poteaux, Adrien Weimann, Martin |
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| CitedBy_id | crossref_primary_10_1007_s00200_024_00669_z crossref_primary_10_3390_math11102324 crossref_primary_10_1016_j_jco_2022_101666 crossref_primary_10_1007_s00037_022_00221_w crossref_primary_10_1007_s10208_024_09646_x |
| Cites_doi | 10.1017/CBO9781139856065 10.1007/BF03167329 10.1145/1390768.1390802 10.1145/1113439.1113457 10.1007/s00208-016-1503-1 10.1016/j.jco.2020.101498 10.1007/BF01178683 10.1145/322063.322068 10.1007/s10208-016-9318-8 10.1007/s00037-013-0063-y 10.1112/S1461157013000089 10.1007/s00200-011-0144-6 10.1145/2755996.2756650 10.1016/j.jsc.2012.05.008 10.1007/BF02242355 10.1006/jcom.1998.0476 10.1090/surv/035 10.1145/2930889.2930931 10.1145/2608628.2608664 10.1007/978-3-0348-5097-1 10.1145/42267.45069 10.1007/3-540-15984-3 10.1145/3055282.3055300 10.1006/jsco.2002.0564 10.1016/0001-8708(89)90009-1 10.1090/surv/006 10.1145/3087604.3087658 10.1016/j.jco.2019.03.002 10.1006/jsco.1994.1025 10.1007/978-1-4612-0265-3 10.1016/j.jsc.2011.08.008 10.1017/CBO9781139171885 10.1145/321879.321890 10.1016/0196-6774(80)90013-9 |
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| Keywords | complexity Puiseux series |
| Language | English |
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