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arXiv 2212.01645math.FA
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曲线 $y=q(x)$ 与 $yx^\ell=1$ 上的截断矩问题

The truncated moment problem on curves $y=q(x)$ and $yx^\ell=1$

Aljaž Zalar

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40 pages; To appear in Linear and Multilinear Algebra
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AI总结：

本文研究两类代数曲线上的二元截断矩问题，通过归约到一元情形改进矩矩阵扩张、平方和次数及最小表示测度原子数上界，并给出线性矩阵不等式判据与若干特例解。

摘要：

本文研究形如 $y=q(x)$（其中 $q(x)\in\mathbb{R}[x]$，$\deg q\geq 3$）以及 $yx^\ell=1$（其中 $\ell\in\mathbb{N}\setminus\{1\}$）的曲线上的二元截断矩问题（TMP）。对于偶数次数序列，Fialkow 首先利用截断 Riesz–Haviland 定理以及在此类曲线上严格为正的多项式的平方和表示，给出了一个基于矩矩阵扩张次数的解。具体而言，该扩张次数的上界关于序列次数和确定曲线的多项式次数是二次的。我们采用归约到一元情形的技术，将 Fialkow 的界改进为：对于曲线 $y=q(x)$ 为 $\deg q-1$，而对于曲线 $yx^\ell=1$ 为 $\ell+1$。这进而使上述平方和表示中的次数得到类似改进。此外，我们得到了最小表示测度中原子数的上界：对于偶数次数序列，曲线 $y=q(x)$ 上为 $k\deg q$，曲线 $yx^\ell=1$ 上为 $k(\ell+1)$；而对于奇数次数序列，曲线 $y=q(x)$ 上为 $k\deg q-\left\lceil\frac{\deg q}{2}\right\rceil$，曲线 $yx^\ell=1$ 上为 $k(\ell+1)-\left\lfloor\frac{\ell}{2}\right\rfloor+1$。在偶数情形下，这些结果与 Riener 和 Schweighofer 的结果相对应，他们对所有平面曲线上的奇数次数序列给出了相同的界；而在奇数情形下，这是在这些特殊情形下对他们界的略微改进。进一步，我们基于与所得一元序列相对应的线性矩阵不等式的可行性，对所研究曲线上的 TMP 给出了另一种解法；最后，我们具体求解了曲线 $y=x^\ell$（$\ell=2,3$）上 TMP 的奇数次数情形，并为曲线 $y=x^2$ 上的偶数次数情形增加了一个新的可解性条件。

英文摘要：

In this paper we study the bivariate truncated moment problem (TMP) on curves of the form $y=q(x)$, $q(x)\in \mathbb{R}[x]$, $\text{deg } q\geq 3$, and $yx^\ell=1$, $\ell\in \mathbb{N}\setminus\{1\}$. For even degree sequences the solution based on the number of moment matrix extensions was first given by Fialkow using the truncated Riesz-Haviland theorem and a sum-of-squares representations for polynomials, strictly positive on such curves. Namely, the upper bound on this number is quadratic in the degrees of the sequence and the polynomial determining a curve. We use a reduction to the univariate setting technique and improve Fialkow's bound to $\text{deg }q-1$ (resp. $\ell+1$) for curves $y=q(x)$ (resp. $yx^\ell=1$). This in turn gives analogous improvements of the degrees in the sum-of-squares representations referred to above. Moreover, we get the upper bounds on the number of atoms in the minimal representing measure, which are $k\;\text{deg }q$ (resp. $k(\ell+1)$) for curves $y=q(x)$ (resp. $yx^\ell=1$) for even degree sequences, while for odd ones they are $k\;\text{deg }q-\big\lceil\frac{\text{deg }q}{2} \big\rceil$ (resp. $k(\ell+1)-\big\lfloor\frac{\ell}{2} \big\rfloor+1$) for curves $y=q(x)$ (resp. $yx^\ell=1$). In the even case these are counterparts to the result by Riener and Schweighofer, which gives the same bound for odd degree sequences on all plane curves, while in the odd case it is a slight improvement of their bound in these special cases. Further on, we give another solution to the TMP on the curves studied based on the feasibility of a linear matrix inequality, corresponding to the univariate sequence obtained, and finally we solve concretely odd degree cases of the TMP on curves $y=x^\ell$, $\ell=2,3$, and add a new solvability condition to the even degree case on the curve $y=x^2$.

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