Waring's problem for pseudo-polynomials

Waring's problem has a long history in additive number theory. In its original form it deals with the representability of every positive integer as sum of $k$-th powers with integer $k$. Instead of these powers we deal with pseudo-polynomials in this paper. A pseudo-polynomial is a ``polynomial'' with at least one exponent not being an integer. Our work extends earlier results on the related problem of Waring for arbitrary real powers $k>12$ by Deshouillers and Arkhipov and Zhitkov.