单项式曲线上的截断矩问题与扩张性质
Truncated Moment Problems and the Extension Property on Monomial Curves
AI总结：
本文确定了由两个单项式间关系定义的平面曲线满足截断矩问题扩张性质的情形，给出了扩张次数界等相关结果。
AI中文摘要：
在多篇论文中，Stochel与Szafraniec从算子理论视角研究代数集上的矩问题，探究满足多项式关系的正定序列何时具有表示测度。在该框架下，Stochel引入A型集，Bisgaard对由两个单项式间关系定义的平面曲线的该性质进行分类。Curto与Fialkow引入了更强的A型性质的截断版本，要求给定次数的半正定扩张的存在性保证表示测度的存在性。受Bisgaard分类的启发，本文确定了哪些由两个单项式间关系定义的平面曲线满足该扩张性质；在肯定情形下，得到了所需扩张次数的显式界；在否定情形下，构造了任意高阶都具有半正定扩张但无支撑在该曲线上的表示测度的截断序列，这些构造得到了在对应曲线上非负但不在其坐标环中为平方和的显式多项式；在肯定情形下，还推导了严格正多项式的平方和证书的显式次数界。
英文摘要：
In several papers, Stochel and Szafraniec studied moment problems on algebraic sets from an operator-theoretic perspective, investigating when positive definite sequences satisfying polynomial relations admit representing measures. Within this framework, Stochel introduced type A sets, and Bisgaard classified the plane curves defined by relations between two monomials that have this property. Curto and Fialkow introduced a stronger, truncated version of the type A property, requiring that the existence of a positive semidefinite extension of prescribed degree guarantees the existence of a representing measure. Motivated by Bisgaard's classification, we determine which plane curves defined by relations between two monomials satisfy this extension property. In the affirmative cases, we obtain explicit bounds on the required extension degree. In the negative cases, we construct truncated sequences that admit positive semidefinite extensions of arbitrarily high order but have no representing measure supported on the curve. These constructions yield explicit polynomials that are nonnegative on the corresponding curves but are not sums of squares in their coordinate rings. In the affirmative cases, we also derive explicit degree bounds for sums-of-squares certificates of strictly positive polynomials.
