A More General Abc Conjecture
نویسندگان
چکیده
In this note we formulate a conjecture generalizing both the abc conjecture of Masser-Oesterlé and the author’s diophantine conjecture for algebraic points of bounded degree. We also show that the latter conjecture implies the new conjecture. As with most of the author’s conjectures, this new conjecture stems from analogies with Nevanlinna theory. In this particular case the conjecture corresponds to relacing the usual counting function of Nevanlinna theory with a truncated counting function. In particular, the abc conjecture of Masser and Oesterlé corresponds to Nevanlinna’s Second Main Theorem with truncated counting functions applied to the divisor [0] + [1] + [∞] on P . The first section of this paper introduces the notation that will be used throughout the paper. Section 2 formulates the new conjecture and discusses some examples related to the new conjecture, including an “ abcde . . . conjecture” and a conjecture of Buium. The third and final section of this paper shows that the new conjecture is implied by the (apparently weaker) older conjecture without truncated counting functions.
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