Positivity Certificates via Integral Representations

多项式矩不等式的平方和证书

Sums of squares certificates for polynomial moment inequalities

本文建立矩多项式代数框架，证明其正定性的平方和证书并构造收敛的半定规划层级，据此解决量子物理中两个非线性 Bell 不等式公开问题。

Positivity and optimization for semi-algebraic functions

We describe algebraic certificates of positivity for functions belonging to a finitely generated algebra of Borel measurable functions, with particular emphasis to algebras generated by semi-algebraic functions. In which case the standard global optimization problem with constraints given by elements of the same algebra is reduced via a natural change of variables to the better understood case of polynomial optimization. A collection of simple examples and numerical experiments complement the theoretical parts of the article.

Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets

This article studies algebraic certificates of positivity for noncommutative (nc) operator-valued polynomials on matrix convex sets, such as the solution set $D_L$, called a free Hilbert spectrahedron, of the linear operator inequality (LOI) $L(X)=A_0\otimes I+\sum_{j=1}^g A_{j}\otimes X_j\succeq 0,$ where $A_j$ are self-adjoint linear operators on a separable Hilbert space, $X_j$ matrices and $I$ is an identity matrix. If $A_j$ are matrices, then $L(X)\succeq 0$ is called a linear matrix inequality (LMI) and $D_L$ a free spectrahedron. For monic LMIs, i.e., $A_0=I$, and nc matrix-valued polyn…

态、迹与矩多项式的正定性，及其在量子信息中的应用

Positivity of state, trace, and moment polynomials, and applications in quantum information

本综述统一介绍了源自量子信息理论的态、迹与矩多项式的正定性与优化，给出了平方和证书，并设计了收敛的半定规划层次结构，最后概述了其在量子信息中的多种应用。

对称函数的任意维Positivstellensätze

Any-dimensional Positivstellensätze for symmetric functions

针对一致有下界的对称函数，证明了无限维的Pólya和Reznick型Positivstellensätze，通过幂和映射与离散概率测度矩的关联刻画无限对称群的轨道空间。

Möbius变换与自函数的积分表示

Möbius transformation and the integral representation of endofunctions

该研究针对Pick函数的Herglotz-Nevanlinna积分表示进行补充，新增Möbius变换公式与边界支撑附近的正性刻画，丰富了相关函数论结果。