  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.
