Quote:
Originally Posted by cmd
Hilbert (10)
L(k,n,x,y,z,...)=R(k,n,x,y,z,...)
no solutions

Are you saying then that the reason why x^10+y^10+z^10 = t^4 does not have any solutions in positive integers is a direct consequence of Matiyasevich's Theorem?
I think that the incorporation of certain ingredients of Matiyasevich's proof and a variant of FLT can be used to prove that x^10+y^10+z^10 = t^4 does not have any solutions in positive integers.