Sharper ABC-based bounds for congruent polynomials

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 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 rad ABC. The second improvement generalizes to m ≥ 3 polynomials A1, . . . , Am of degree at most ((3m − 5)/(3m − 7)) deg h − (6/(3m − 7)m) deg radA1 · · ·Am. Manuscrit reçu le 2004.02.10. Permanent ID of this document: 1d9e079cee20138de8e119a99044baa3. 2000 Mathematics Subject Classification. Primary 11C08. The author was supported by the National Science Foundation under grant DMS–0140542, and by the Alfred P. Sloan Foundation. He used the libraries at the Mathematical Sciences Research Institute and the University of California at Berkeley. 2 Daniel J. Bernstein