On polynomial representation by U-numbers

Let n be a positive integer and let \(P(x,y)\in \mathbb {Z}[x,y]\) $P(x,y)\in \mathbb{Z}[x,y]$ be a non-constant polynomial. In this paper, we prove that every S- and T-number (under some technical conditions) can be written in the form \(P(\sigma , \tau )\) $P(\sigma ,\tau )$ for uncountable many pairs \((\sigma , \tau )\) $(\sigma ,\tau )$ of \(U_n\) $U_n$ -numbers.