论文标题
曲线上的截断力矩问题$ y = q（x）$和$ yx^\ ell = 1 $
The truncated moment problem on curves $y=q(x)$ and $yx^\ell=1$
论文作者
论文摘要
在本文中，我们研究了$ y = q（x）$，$ q（x）\ in \ mathbb {r} [r} [x] $，$ \ text {deg} q \ geq 3 $和$ yx^\ ell = 1 $，$ \ ell \ in \ mathbb {r} [x] $，$ y = q（x）$，$ y = q（x）$，$ y = q（x）\ in curves的双变量截短力矩问题（TMP） \ mathbb {n} \ setMinus \ {1 \} $。对于偶数序列，基于矩矩阵扩展的数量，Fialkow首先使用截短的Riesz-Haviland定理和多项式的平方总和表示，首先是在此类曲线上进行的。也就是说，该数字上的上限在序列的程度和多项式确定曲线的程度上是二次的。我们对单变量设置技术进行了简化，并改善了Fialkow的绑定到$ \ text {deg} q-1 $（curves $ y = q（x）$（suves。yx^\ ell = 1 $）。反过来，这在上面提到的平方表示表示总和表示中提供了类似的改进。 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 $（rep. $ k（\ ell+1） - \ big \ big \ lfloor \ lfloor \ frac {\ ell} {2} {2} {2} {2} \ big big \ rfloor+1 $ 1 $ $ y $ y y = $ yx^\ ell = 1 $）。在偶数情况下，这些都是Riener和Schweighofer结果的对应物，在所有平面曲线上，这给出了奇数序列相同的结合，而在奇数情况下，在这些特殊情况下，它们的结合略有改善。此外，我们根据线性矩阵不平等的可行性，在研究的曲线上为TMP提供了另一种解决方案，这对应于获得的单变量序列，最后我们解决了TMP曲线上TMP的奇数奇数案例$ y = y = x^\ ell $，$ \ ell $，$ \ ell = 2,3 $，并添加了一个新的求解情况。
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 $y=x^2$.
