标题： 正多项式与平面上三次曲线的截断矩问题 显示英文标题

标题： Positive polynomials and the truncated moment problem on plane cubics

摘要： 截断矩问题在给定的闭集$K$上在$\mathbb{R}^2$中（$K$-TMP）要求表征一个给定的双变量有界次数多项式的线性泛函具有相对于博雷尔测度$\mu$的积分表示的条件，其中$\text{supp}\mu\subseteq K$。 $K$-TMP 的解已知于$K$，这是一条直线、一条二次曲线以及一些三次曲线的情况。 在本文中，我们解决了每个三次曲线$C$的$C$-TMP。 我们的第一个结果指出，有界次数多项式锥体中在$C$上非负的极端射线仅具有实根。 这个结果使我们能够建立在$C$上多项式的正性证书，并带有次数限制。 为了获得这些证书的具体形式，需要进行逐个情况的分析。 在仿射线性变量变换下，我们将三次曲线分为29种情况，其中13种是不可约的，16种是可约的。 利用这些证书，我们具体地根据两个或三个局部化矩阵地半正定性来解决非奇异的$C$-TMP。 在大多数不可约情况下，我们也提供了非奇异和奇异的$C$-TMP的构造性解，这些解可用于具体计算表示测度。 还获得了Carathéodory数的上界，在某些情况下这些上界是精确的，或者最多与精确界相差1。 显示英文摘要

摘要： The truncated moment problem supported on a given closed set $K$ in $\mathbb{R}^2$ ($K$-TMP) asks to characterize conditions for a given linear functional on bivariate polynomials of bounded degree to have an integral representation with respect to a Borel measure $\mu$ with $\text{supp}\mu\subseteq K$. The solutions to the $K$-TMP are known for $K$, which is a line, a quadratic curve, and for some cases of cubic curves. In this paper, we solve the $C$-TMP for every cubic curve $C$. Our first result states that the extreme rays of the cone of polynomials of bounded degree, nonnegative on $C$, have only real zeroes. This result allows us to establish certificates for the positivity of polynomials on $C$ with degree bounds. To obtain concrete forms of these certificates, a case-by-case analysis is required. Up to affine linear change of variables, we divide cubics into 29 cases, 13 irreducible and 16 reducible ones. Using the certificates, we concretely solve the nonsingular $C$-TMP in terms of positive semidefiniteness of two or three localizing moment matrices. In most irreducible cases, we also provide constructive solutions to the nonsingular and singular $C$-TMPs, which can be used to compute a representing measure concretely. Upper bounds for the Carath\'eodory number are also obtained, which in some cases are sharp or differ by at most 1 from the sharp bound.