  Given $d,s \in \mathbb{N}$, a finite set $A \subseteq \mathbb{Z}$ and
polynomials $\varphi_1, \dots, \varphi_{s} \in \mathbb{Z}[x]$ such that $1 \leq
deg \varphi_i \leq d$ for every $1 \leq i \leq s$, we prove that \[ |A^{(s)}| +
|\varphi_1(A) + \dots + \varphi_s(A) | \gg_{s,d} |A|^{\eta_s} , \]
  for some $\eta_s \gg_{d} \log s / \log \log s$. Moreover if $\varphi_i(0)
\neq 0$ for every $1 \leq i \leq s$, then \[ |A^{(s)}| + |\varphi_1(A) \dots
\varphi_s(A) | \gg_{s,d} |A|^{\eta_s}. \] These generalise and strengthen
previous results of Bourgain--Chang, P\'{a}lv\"{o}lgyi--Zhelezov and
Hanson--Roche-Newton--Zhelezov. We derive these estimates by proving the
corresponding low-energy decompositions. The latter furnish further
applications to various problems of a sum-product flavour, including questions
concerning large additive and multiplicative Sidon sets in arbitrary sets of
integers.
