Value distributions for read-once polynomials on finite fields

Abstract

Consider read-once polynomial functions over a finite field of order $k$, i.e., functions defined by expressions built from field addition, multiplication, and constants, in which every variable occurs at most once. Let $p=(p_1,\dots,p_k)$ be the distribution of the values of a read-once function on independent uniform inputs. We prove that for every $k\ge4$ all such distributions belong to a body $\mathcal{B}_k$, defined by the following relation on the sorted atoms $p_1^\downarrow\geq\cdots\geq p_k^\downarrow$ of the distribution: \[ \mathcal{B}_k=\left\{p:p_k^\downarrow\ge \frac{1-p_2^\downarrow-(1-p_2^\downarrow)^k}{k-1}\right\}, \] or equivalently, $\mathcal{B}_k = \{ p \colon 1-p_2^\downarrow-(k-1)p_k^\downarrow\le(1-p_2^\downarrow)^k\}$. Our main theorem is that this body is stable under the convolutions corresponding to both field operations. More generally, convolution for any quasigroup operation on $k$ points preserves $\mathcal{B}_k$; the multiplicative conclusion needs only an absorbing zero and a quasigroup operation on the nonzero elements. The body is full-dimensional and contains the uniform law and every point mass; its normalized volume is given by an exact one-dimensional integral, and the $k$th root of that volume tends to $0.2183305369\ldots$. The complete development, including the two stability theorems, the exact volume formula, and its sharp exponential rate, has been formalized in Lean 4.