Activity: Talk or presentation › Talk at conference or symposium › Science to science
In this talk I will discuss several polynomial variants of some well-known Pillai's Diophantine problems. In particular, in my focus is the equation a1p(x)n1+b1q(x)m1=a2p(x)n2+b2q(x)m2=f(x) in complex polynomials f,p,q with f nonzero and p and q nonconstant, nonzero complex numbers ai,bi and positive integers ni,mi. Of particular importance is the special case a1=a2=1 and b1=b2=−1 corresponding to a well-studied Pillai's Diophantine equation an1−bm1=an2−bm2 in positive integers ni,mi,a,b with a>1 and b>1. Furthermore, I will show that for nonconstant coprime complex polynomials p and q, the number of solutions (n,m)∈N2 of 0≤deg(p(x)n−q(x)m)≤d is asymptotically equal to d2/(degpdegq) as d→∞, as well as consider a generalization of this problem where the powers of polynomials are replaced by the sums of powers of polynomials.