Divyum Sharma, Department of Pure Mathematics, University of Waterloo
“Number of solutions of Thue Equations”
LetF(X,Y)∈Z[X,Y]beabinaryformwithnon-zerodiscriminantanddegreer≥3. Lethbea non-zero integer. In 1909, Thue proved that the equation F (X, Y ) = h has only finitely many solutions in integers x and y. Since then, much work has been done to estimate the number of solutions of Thue equations from above. In this talk, we will consider forms F which are diagonalizable, i.e. forms which can be written as F (X, Y ) = (αX + βY )r − (γX + δY )r, where the constants α, β, γ and δ are such that F (X, Y ) ∈ Z[X, Y ] and αδ − βγ ̸= 0.
Following the hypergeometric method of Thue and Siegel, we give upper bounds for the number of primitive solutions of the Thue inequality 0 < |F (X, Y )| ≤ h, where F is diagonalizable. This is joint work with S. Akhtari and N. Saradha.
MC 5479