There is no polynomial formula for the catenary and the tame degree of finitely generated monoids

In the last two decades there has been a wealth of results determining the precise value of the catenary degree and the tame degree. Mostly, however, only for very special classes of monoids and domains. In the present work we now show that there is no polynomial formula, neither for the catenary nor for the tame degree, which is valid for a sufficiently large class of finitely generated monoids.