Elimination ideal and bivariate resultant over finite fields
17 pages International audience A new algorithm is presented for computing the largest degree invariant factor of the Sylvester matrix (with respect either to $x$ or $y$) associated to two polynomials $a$ and $b$ in $\mathbb F_q[x,y]$ which have no non-trivial common divisors. The algorithm is rando...
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ftccsdartic:oai:HAL:hal-03999414v1 2023-12-03T10:31:18+01:00 Elimination ideal and bivariate resultant over finite fields Villard, Gilles Centre National de la Recherche Scientifique (CNRS) Laboratoire de l'Informatique du Parallélisme (LIP) École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL) Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS) Arithmétiques des ordinateurs, méthodes formelles, génération de code (ARIC) Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL) Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Inria Lyon Institut National de Recherche en Informatique et en Automatique (Inria) Tromsø Norway, Norway 2023-07-24 https://hal.science/hal-03999414 https://hal.science/hal-03999414/document https://hal.science/hal-03999414/file/ms.pdf https://doi.org/10.1145/3597066.3597100 en eng HAL CCSD ACM info:eu-repo/semantics/altIdentifier/arxiv/2302.08891 info:eu-repo/semantics/altIdentifier/doi/10.1145/3597066.3597100 hal-03999414 https://hal.science/hal-03999414 https://hal.science/hal-03999414/document https://hal.science/hal-03999414/file/ms.pdf ARXIV: 2302.08891 doi:10.1145/3597066.3597100 http://creativecommons.org/licenses/by/ info:eu-repo/semantics/OpenAccess Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation ISSAC 2023: International Symposium on Symbolic and Algebraic Computation 2023 https://hal.science/hal-03999414 ISSAC 2023: International Symposium on Symbolic and Algebraic Computation 2023, Jul 2023, Tromsø Norway, Norway. pp.526-534, ⟨10.1145/3597066.3597100⟩ [INFO]Computer Science [cs] info:eu-repo/semantics/conferenceObject Conference papers 2023 ftccsdartic https://doi.org/10.1145/3597066.3597100 2023-11-04T23:38:56Z 17 pages International audience A new algorithm is presented for computing the largest degree invariant factor of the Sylvester matrix (with respect either to $x$ or $y$) associated to two polynomials $a$ and $b$ in $\mathbb F_q[x,y]$ which have no non-trivial common divisors. The algorithm is randomized of the Monte Carlo type and requires $O((de)^{1+\epsilon}\log(q) ^{1+o(1)})$ bit operations, where $d$ an $e$ respectively bound the input degrees in $x$ and in $y$. It follows that the same complexity estimate is valid for computing: a generator of the elimination ideal $\langle a,b \rangle \cap \mathbb F_q[x]$ (or $\mathbb F_q[y]$), as soon as the polynomial system $a=b=0$ has not roots at infinity; the resultant of $a$ and $b$ when they are sufficiently generic, especially so that the Sylvester matrix has a unique non-trivial invariant factor. Our approach is to use the reduction of the problem to a problem of minimal polynomial in the quotient algebra $\mathbb F_q[x,y]/\langle a,b \rangle$. By proposing a new method based on structured polynomial matrix division for computing with the elements in the quotient, we manage to improve the best known complexity bounds. Conference Object Tromsø Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) Norway Tromsø Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation 526 534 |
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[INFO]Computer Science [cs] |
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[INFO]Computer Science [cs] Villard, Gilles Elimination ideal and bivariate resultant over finite fields |
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[INFO]Computer Science [cs] |
description |
17 pages International audience A new algorithm is presented for computing the largest degree invariant factor of the Sylvester matrix (with respect either to $x$ or $y$) associated to two polynomials $a$ and $b$ in $\mathbb F_q[x,y]$ which have no non-trivial common divisors. The algorithm is randomized of the Monte Carlo type and requires $O((de)^{1+\epsilon}\log(q) ^{1+o(1)})$ bit operations, where $d$ an $e$ respectively bound the input degrees in $x$ and in $y$. It follows that the same complexity estimate is valid for computing: a generator of the elimination ideal $\langle a,b \rangle \cap \mathbb F_q[x]$ (or $\mathbb F_q[y]$), as soon as the polynomial system $a=b=0$ has not roots at infinity; the resultant of $a$ and $b$ when they are sufficiently generic, especially so that the Sylvester matrix has a unique non-trivial invariant factor. Our approach is to use the reduction of the problem to a problem of minimal polynomial in the quotient algebra $\mathbb F_q[x,y]/\langle a,b \rangle$. By proposing a new method based on structured polynomial matrix division for computing with the elements in the quotient, we manage to improve the best known complexity bounds. |
author2 |
Centre National de la Recherche Scientifique (CNRS) Laboratoire de l'Informatique du Parallélisme (LIP) École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL) Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS) Arithmétiques des ordinateurs, méthodes formelles, génération de code (ARIC) Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL) Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Inria Lyon Institut National de Recherche en Informatique et en Automatique (Inria) |
format |
Conference Object |
author |
Villard, Gilles |
author_facet |
Villard, Gilles |
author_sort |
Villard, Gilles |
title |
Elimination ideal and bivariate resultant over finite fields |
title_short |
Elimination ideal and bivariate resultant over finite fields |
title_full |
Elimination ideal and bivariate resultant over finite fields |
title_fullStr |
Elimination ideal and bivariate resultant over finite fields |
title_full_unstemmed |
Elimination ideal and bivariate resultant over finite fields |
title_sort |
elimination ideal and bivariate resultant over finite fields |
publisher |
HAL CCSD |
publishDate |
2023 |
url |
https://hal.science/hal-03999414 https://hal.science/hal-03999414/document https://hal.science/hal-03999414/file/ms.pdf https://doi.org/10.1145/3597066.3597100 |
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Tromsø Norway, Norway |
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Norway Tromsø |
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Norway Tromsø |
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Tromsø |
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Tromsø |
op_source |
Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation ISSAC 2023: International Symposium on Symbolic and Algebraic Computation 2023 https://hal.science/hal-03999414 ISSAC 2023: International Symposium on Symbolic and Algebraic Computation 2023, Jul 2023, Tromsø Norway, Norway. pp.526-534, ⟨10.1145/3597066.3597100⟩ |
op_relation |
info:eu-repo/semantics/altIdentifier/arxiv/2302.08891 info:eu-repo/semantics/altIdentifier/doi/10.1145/3597066.3597100 hal-03999414 https://hal.science/hal-03999414 https://hal.science/hal-03999414/document https://hal.science/hal-03999414/file/ms.pdf ARXIV: 2302.08891 doi:10.1145/3597066.3597100 |
op_rights |
http://creativecommons.org/licenses/by/ info:eu-repo/semantics/OpenAccess |
op_doi |
https://doi.org/10.1145/3597066.3597100 |
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Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation |
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526 |
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534 |
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