发表机构
Max Planck Institute for Mathematics in the Sciences; Korea Advanced Institute of Science and Technology (KAIST)（马克斯·普朗克数学科学研究所; 韩国科学技术院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过射影曲线刚性和矩阵方法，证明有限自由Stam不等式中Hermite多项式为唯一极值元，并刻画了曲线分裂与碰撞结构。
AI 中文摘要
我们刻画了有限自由Stam不等式和熵幂不等式中的等号成立条件，证明了在简单实根输入中，Hermite多项式是唯一的极值元（在独立的平移和缩放意义下）。证明将该分类问题转化为射影平面曲线的刚性问题。利用双曲性和Helton-Vinnikov定理，我们将Jacobian缺陷表示为确定对称束中的非对角平方范数。结合分数传输，这给出了对所有实根输入下Stam不等式的矩阵证明。对于每个简单基对，缺陷消失的方向正是独立的平移和共同的膨胀，在每个次数中形成一个三维子空间，其曲线分裂为$n$条射影直线。在碰撞处，领先构型再次是碰撞输入簇的速度多项式的归一化导数的有限自由卷积。结合关联计数，该局部公式表明至多一个实纤维是奇异的，每个实奇点都是普通全实多重点，且有序碰撞重数决定了实归一化覆盖。对于$n\geq3$，每条非分裂曲线至少有$2n-2$个非实射影判别式零点（按重数计），等号由通过每个简单输入对的几何亏格为零的不可约曲线取得。领先的Fisher信息系数由碰撞切线构型决定，而有限熵项保留簇之间的间隙。对于$n\geq3$，沿最优加权分数方向的最大对数熵散度等价于Stam等号成立。
英文摘要
We characterize equality in the finite free Stam and entropy-power inequalities, proving that Hermite polynomials are the unique extremizers among simple real-rooted inputs, up to independent translations and scalings. The proof turns this classification into a rigidity problem for projective plane curves. Using hyperbolicity and the Helton-Vinnikov theorem, we express the Jacobian defect as an off-diagonal squared norm in a definite symmetric pencil. Together with score transport, this yields a matrix proof of Stam for all real-rooted inputs. For each simple base pair, the directions in which the defect vanishes are the independent translations and common dilation, forming a three-dimensional subspace in every degree whose curves split into $n$ projective lines. At a collision, the leading configurations are again finite free convolutions of normalized derivatives of the velocity polynomials of the colliding input clusters. Combined with incidence counting, this local formula shows that at most one real fiber is singular, every real singularity is an ordinary totally real multiple point, and the ordered collision multiplicities determine the real normalization covering. For $n\geq3$, every non-split curve has at least $2n-2$ non-real projective discriminant zeros, counted with multiplicity, with equality attained by irreducible curves of geometric genus zero through every simple input pair. The leading Fisher-information coefficient is determined by the colliding tangent configurations, while the finite entropy term retains the gaps between clusters. For $n\geq3$, maximal logarithmic entropy divergence along the optimally weighted score direction is equivalent to Stam equality.
