Sharper ABC-based bounds for congruent polynomials

Daniel J. Bernstein[1]

  • [1] Department of Mathematics, Statistics, and Computer Science (M/C 249) The University of Illinois at Chicago Chicago, IL 60607–7045

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 3, page 721-725
  • ISSN: 1246-7405

Abstract

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Agrawal, Kayal, and Saxena recently introduced a new method of proving that an integer is prime. The speed of the Agrawal-Kayal-Saxena method depends on proven lower bounds for the size of the multiplicative semigroup generated by several polynomials modulo another polynomial h . Voloch pointed out an application of the Stothers-Mason ABC theorem in this context: under mild assumptions, distinct polynomials A , B , C of degree at most 1 . 2 deg h - 0 . 2 deg rad A B C cannot all be congruent modulo h . This paper presents two improvements in the combinatorial part of Voloch’s argument. The first improvement moves the degree bound up to 2 deg h - deg rad A B C . The second improvement generalizes to m 3 polynomials A 1 , , A m of degree at most ( ( 3 m - 5 ) / ( 3 m - 7 ) ) deg h - ( 6 / ( 3 m - 7 ) m ) deg rad A 1 A m .

How to cite

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Bernstein, Daniel J.. "Sharper ABC-based bounds for congruent polynomials." Journal de Théorie des Nombres de Bordeaux 17.3 (2005): 721-725. <http://eudml.org/doc/249452>.

@article{Bernstein2005,
abstract = {Agrawal, Kayal, and Saxena recently introduced a new method of proving that an integer is prime. The speed of the Agrawal-Kayal-Saxena method depends on proven lower bounds for the size of the multiplicative semigroup generated by several polynomials modulo another polynomial $h$. Voloch pointed out an application of the Stothers-Mason ABC theorem in this context: under mild assumptions, distinct polynomials $A,B,C$ of degree at most $1.2 \deg h - 0.2 \deg \operatorname\{rad\}ABC$ cannot all be congruent modulo $h$. This paper presents two improvements in the combinatorial part of Voloch’s argument. The first improvement moves the degree bound up to $2 \deg h - \deg \operatorname\{rad\}ABC$. The second improvement generalizes to $m\ge 3$ polynomials $A_1,\dots ,A_m$ of degree at most $((3m-5)/(3m-7)) \deg h - (6/(3m-7)m) \deg \operatorname\{rad\}A_1\cdots A_m$.},
affiliation = {Department of Mathematics, Statistics, and Computer Science (M/C 249) The University of Illinois at Chicago Chicago, IL 60607–7045},
author = {Bernstein, Daniel J.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Stothers-Mason ABC-theorem},
language = {eng},
number = {3},
pages = {721-725},
publisher = {Université Bordeaux 1},
title = {Sharper ABC-based bounds for congruent polynomials},
url = {http://eudml.org/doc/249452},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Bernstein, Daniel J.
TI - Sharper ABC-based bounds for congruent polynomials
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 3
SP - 721
EP - 725
AB - Agrawal, Kayal, and Saxena recently introduced a new method of proving that an integer is prime. The speed of the Agrawal-Kayal-Saxena method depends on proven lower bounds for the size of the multiplicative semigroup generated by several polynomials modulo another polynomial $h$. Voloch pointed out an application of the Stothers-Mason ABC theorem in this context: under mild assumptions, distinct polynomials $A,B,C$ of degree at most $1.2 \deg h - 0.2 \deg \operatorname{rad}ABC$ cannot all be congruent modulo $h$. This paper presents two improvements in the combinatorial part of Voloch’s argument. The first improvement moves the degree bound up to $2 \deg h - \deg \operatorname{rad}ABC$. The second improvement generalizes to $m\ge 3$ polynomials $A_1,\dots ,A_m$ of degree at most $((3m-5)/(3m-7)) \deg h - (6/(3m-7)m) \deg \operatorname{rad}A_1\cdots A_m$.
LA - eng
KW - Stothers-Mason ABC-theorem
UR - http://eudml.org/doc/249452
ER -

References

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  1. Daniel J. Bernstein, Proving primality in essentially quartic random time. Mathematics of Computation, to appear. http://cr.yp.to/papers.html##quartic Zbl1144.11085MR2261028
  2. Noah Snyder, An alternate proof of Mason’s theorem. Elemente der Mathematik 55 (2000), 93–94. http://www.springerlink.com/openurl.asp?genre=article&issn=0013-6018&volume=55&issue=3&spage=93 Zbl1031.11012MR1781918
  3. José Felipe Voloch, On some subgroups of the multiplicative group of finite rings. Journal de Théorie des Nombres de Bordeaux 16 (2004), 1246–7405. http://www.ma.utexas.edu/users/voloch/preprint.html Zbl1078.11069MR2145584

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