Let F(x, y) be an irreducible binary form of degree at least three with integer co-efficients. We consider the problem of bounding the number of integer solutions (p, q) to the Thue equation |F(x, y)| = 1. Our particular area of study is the case where F(x, y ) is a tetranomial, that is, it has exactly four non-zero coefficients. Building on ideas developed by Thomas in his work on three-term Thue equations, we prove our main result:;Theorem. Let F(x, y) be an irreducible binary form given by F&parl0;x,y&parr0;=a0xn+r0 xmyn-m-s0xky n-k+t0yn, with n > m > k > 0, a0 ∈ Z+, and r0, s0, t 0 ∈ Z {0}, such that n ≥ 50 and for C = 0.99, Ca0n > |r0|m and C|t0|n > | s0|(n -- k). Then, the equation |F(x, y)| = 1 has at most 36 solutions (p, q) ∈ Z2 with |pq| ≥ 2 (where (p, q) and ( --p, --q) are counted as a single solution). Moreover, if n is odd, then there are at most 30 such solutions .;Our proofs use a combination of classical arguments, later developed by Bombieri, Mueller, and Schmidt, and more recent methods used by Thomas. More specifically, we apply the Thue-Siegel principle and a strong gap principle to bound the number of large solutions, and we adapt Thomas's approach, which involves solving a Diophantine approximation problem and using another gap principle, to bound the number of small solutions. Throughout, we develop additional techniques needed for dealing with the distinct complexities inherent in working with tetranomial Thue equations.
展开▼
