Sharper ABC-based bounds for congruent polynomials

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 -