AI 中文总结
本文研究自由代数中对称非交换多项式的有理平方和，给出其算子理论刻画，证明厄米特平方和为其真子集（两生成元时），且该集合及其补集在三生成元时均非凸。
AI 中文摘要
本文针对自由代数中的对称非交换多项式，引入了有理平方和的概念。在假设该多项式存在严格正的标量自伴赋值的前提下，本文的主要结果对该类多项式给出了算子理论刻画：一个多项式为有理平方和，当且仅当存在一个次数界，使得对于每个满足自然非退化条件的自伴算子赋值，该赋值多项式在对应的有限维循环子空间上的压缩不是负定的。本文还证明，当自由代数至少有两个生成元时，厄米特平方和构成有理平方和的真子集；而对于齐次多项式，这两类集合是重合的。最后，本文表明，当生成元至少为三个时，有理平方和的集合是非凸的，且其补集也非凸。
英文摘要
This paper introduces rational sums of squares for symmetric noncommutative polynomials in free algebras. Under the assumption that the polynomial admits a strictly positive scalar self-adjoint evaluation, our main result gives an operator-theoretic characterization of this class: a polynomial is a rational sum of squares if and only if there exists a degree bound such that, for every self-adjoint operator evaluation satisfying a natural nondegeneracy condition, the compression of the evaluated polynomial to the associated finite-dimensional cyclic subspace is not negative definite. We also prove that sums of Hermitian squares form a proper subset of rational sums of squares when the free algebra has at least two generators, whereas the two classes coincide for homogeneous polynomials. Finally, we show that the set of rational sums of squares is nonconvex when there are at least three generators, and that its complement is also nonconvex.
