# The multidimensional truncated Moment Problem: Carath\'eodory Numbers from Hilbert Functions

28 Jun 2019 di Dio Philipp J. Kummer Mario

In this paper we improve the bounds for the Carath\'eodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We find that for every $\varepsilon>0$ and $d\in\mathbb{N}$ there is a $n\in\mathbb{N}$ such that we can construct a moment functional $L:\mathbb{R}[x_1,\dots,x_n]_{\leq d}\rightarrow\mathbb{R}$ which needs at least $(1-\varepsilon)\cdot\left(\begin{smallmatrix} n+d\\ n\end{smallmatrix}\right)$ atoms $l_{x_i}$... (read more)

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• FUNCTIONAL ANALYSIS