  Maximally supersymmetric field theories in various dimensions are believed to
possess special properties due to extended supersymmetry. In four dimensions
they are free from UV divergences but are IR divergent on shell, in higher
dimensions, on the contrary, they are IR finite but UV divergent. In what
follows we consider the four-point on-shell scattering amplitudes in D=6,8,10
supersymmetric Yang-Mills theory in the planar limit within the spinor-helicity
and on shell supersymmetric formalism. We study the UV divergences and
demonstrate how one can sum them over all orders of PT. Analyzing the
R-operation we obtain the recursive relations and derive differential equations
that sum all leading, subleading, etc., divergences in all loops generalizing
the standard RG formalism for the case of nonrenormalizable interactions. We
then perform the renormalization procedure which differs from the ordinary one
in that the renormalization constant becomes the operator depending on
kinematics. Solving the obtained RG equations for particular sets of diagrams
analytically and for the general case numerically, we analyze their high energy
behaviour and find out that while each term of PT increases as a power of
energy the total sum behaves differently: in D=6 two partial amplitudes
decrease with energy and the third one increases exponentially, while in D=8
and 10 the amplitudes possess an infinite number of periodic poles at finite
energy.
