Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{i\in S}x_i\) be a real multilinear polynomial of degree at most \(k\); let \(I_i(F)=\sum _{S\ni i}a_S^2\). Let \((X_i)\) be independent real inputs on a probability space, each measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). Then \(\mathbb E[F(X)^4]\le 9^k(\sum _S a_S^2)^2\).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{j\in S}x_j\) be a real multilinear polynomial and fix \(i\in \{ 1,\ldots ,n\} \). Define \(A_i(x)=\sum _{S:i\notin S}a_S\prod _{j\in S}x_j\) and \(D_i(x)=\sum _{S:i\notin S}a_{S\cup \{ i\} }\prod _{j\in S}x_j\). Suppose \(F\) has degree at most \(k+1\) and \((X_j)\) are independent real inputs on a probability space, each measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). Then \(\mathbb E[D_i(X)^4]\le 9^k I_i(F)^2\), where \(I_i(F)=\sum _{S\ni i}a_S^2\).