On the linearity of Artin functions

It was proved by Elkik that, under some smoothness conditions, the Artin functions of systems of polynomials over a Henselian pair are bounded above by linear functions. This paper gives a stronger form of this result for the class of excellent Henselian local rings. The linearity of Artin functions of systems of polynomials in one variable is also studied. Explicit calculations of Artin functions of monomials and determinantal ideals are also included.