Let $X$ be a discrete random variable with possible values $\{x_1,\ldots,x_n\}$, and let $p$ denote the probability mass function of $X$. In addition, denote $p_i=p(x_i)$.

The entropy of $X$ is defined as follows $$H(X)=H(p_1,\ldots,p_n)=-\sum_{i=1}^n p_i\log p_i$$

I'm looking for a low degree (up to $\log n$) polynomial $P(p_1,\ldots,p_n)$ which provides as good as possible approximation for the entropy of the distribution.