We give a generalization of the Shestakov-Umirbaev inequality which plays an important role in their solution of the Tame Generators Problem on the automorphism group of a polynomial ring. As an application, we give a new necessary condition for endomorphisms of a polynomial ring to be invertible, which implies Jung's theorem in case of two variables.
Shigeru KURODA. "A generalization of the Shestakov-Umirbaev inequality." J. Math. Soc. Japan 60 (2) 495 - 510, April, 2008. https://doi.org/10.2969/jmsj/06020495