有限域上只读多项式的值分布
Value distributions for read-once polynomials on finite fields
AI总结：
该研究确定k≥4时有限域上只读多项式的值分布所属的体ℬ_k，证明其对卷积稳定，给出其体积公式及极限，相关结论已在Lean 4中形式化。
AI中文摘要：
考虑有限域GF(k)上的只读多项式函数，即由域加法、乘法和常数构造的表达式定义的函数，其中每个变量最多出现一次。设p=(p₁,…,p_k)为只读函数在独立均匀输入下的值分布，我们证明：对所有k≥4，这类分布均属于体ℬ_k，该体由分布的排序原子p₁^↓≥…≥p_k^↓满足的关系定义：ℬ_k={p:p_k^↓≥(1-p₂^↓-(1-p₂^↓)^k)/(k-1)}，等价于ℬ_k={p:1-p₂^↓-(k-1)p_k^↓≤(1-p₂^↓)^k}。我们的主定理表明，该体对对应两种域运算的卷积是稳定的；更一般地，k个元素上的任意拟群运算的卷积都保持ℬ_k，乘法结论仅需一个吸收零元和非零元上的拟群运算。该体是满维的，包含均匀分布和所有单点分布；其归一化体积由精确的一维积分给出，该体积的k次方根趋于0.2183305369…。完整推导（含两个稳定性定理、精确体积公式及其尖锐指数速率）已在Lean 4中形式化。
英文摘要：
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.
