The Algebraic Limits of Polynomial Information Measures

The paper studies the existence of \emph{polynomial} measures of dependence between two random variables: polynomial functions of the joint distribution that (i) vanish on independence and (ii) cannot increase under post-processing of either variable (the data processing inequality, DPI). Mutual information satisfies both properties but is transcendental in the joint distribution, making it impossible to estimate without bias from finitely many samples. A polynomial alternative would admit an exact finite-sample unbiased estimator, with the polynomial degree controlling the required sample size. The main result is negative: in the asymmetric setting where the variables have different alphabet sizes (\(|X| > |Y|\)), no nonzero polynomial can simultaneously satisfy DPI on the larger-alphabet side and vanish on independence. In the symmetric case \(|X| = |Y| = n\), we establish a structural result: every such polynomial is divisible by \((\det U)^2\), where \(U\) denotes the \(n \times n\) joint distribution matrix. Consequently, any nontrivial candidate must have degree at least \(2n\). The determinant-based mutual information attains this lower bound. These algebraic results have direct consequences for \emph{multi-task peer prediction}, a mechanism-design problem in which a principal incentivizes honest reports from agents who observe correlated signals but no ground truth. Every such mechanism running on \(\ell\) independent tasks gives rise to a polynomial measure of dependence of degree at most \(\ell\), so our lower bounds on polynomial degree translate directly into lower bounds on the number of tasks: in the asymmetric case no finite-task mechanism exists on the larger-alphabet side at all, while the symmetric case requires at least \(\ell \geq 2n\) tasks.