Date & Time: Wednesday, January 15, 2014, 16:00-17:00.
Venue: Ramanujan Hall
Speaker: N. Saradha, TIFR Mumbai
Title: Number of representations of integers by binary forms
Abstract: Let F(x,y) be an irreducible binary form of degree r >= 3 with integer coefficients and h a non-zero integer. In a seminal work in 1909, Thue proved that the equation F(x,y) = h has only fnitely many solutions in integers x and y. For this purpose, he employed a method based on approximation of algebraic numbers by rationals. His method was developed by several mathematicians to give better estimates for the number of solutions of Thue equations. In 1987, Bombieri and Schmidt estimated the number of primitive solutions as cr1+w(h), where c is an absolute constant and w(h) denotes the number of distinct prime divisors of h. Further they showed that c = 215 if r >= r0 where r0 is unspecified. In a joint work with Divyum Sharma, we showed that the above result of Bombieri and Schmidt is true with r0 = 23. Further, c may be taken as small as 10 provided the discriminant of F is large compared to r. In this talk, I shall indicate some salient features of Thue's method and how the improvement in our work has been obtained.