Tame roots of wild quivers

We study the tame behaviour of the representations of wild quivers Q via tame roots. A positive root d of Q is called a tame root if d is sincere and for any positive sub-root d^′ of d we have q(d^′)0, where q(d^′) is the Tits form of Q. We prove that a sincere root is a tame root if and only if for any decomposition of d into a sum of positive sub-roots d=d¹++d^s, there is at most one dⁱ with q(dⁱ)=0 and q(d^j)=1. This is the essential property of a tame root and it is an alternative way to define tame roots. Then we give the canonical decomposition of a tame root. At the end we prove our main result that for any wild graph, there are only finitely many tame roots.