On a problem of diophantus with polynomials

Let m >= 2 and k >= 2 be integers, and let R be a commutative ring with a unit element denoted by 1. A kth power diophantine m-tuple in R is an m-tuple (alpha(1), alpha(2), . . . , alpha(m)) of nonzero elements of R such that alpha(i)alpha(j) + 1 is a kth power of an element of R for 1 <= i < j <= m. In this paper, we investigate the case when k >= 3 and R = K [X], the ring of polynomials with coefficients in a field K of characteristic zero. We prove the following upper bounds on m. the size of diophantine m-tuple: m <= 5 if k = 3; m <= 4 if k = 4; m <= 3 for k >= 5; m <= 2 for k even and k >= 8.