In this paper we improve the bounds for the Carathéodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, \(\mathbb {R}^n\), and \([0,1]^n\). We also treat moment problems with small gaps. 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 \(L:\mathbb {R}[x_1,\cdots ,x_n]_{\le d}\rightarrow \mathbb {R}\) which needs at least \((1-\varepsilon )\cdot \left( {\begin{matrix} n+d\\ n\end{matrix}}\right) \) atoms \(l_{x_i}\). Consequences and results for the Hankel matrix and flat extension are gained. We find that there are moment functionals \(L:\mathbb {R}[x_1,\cdots ,x_n]_{\le 2d}\rightarrow \mathbb {R}\) which need to be extended to the worst case degree 4d, \(\tilde{L}:\mathbb {R}[x_1,\cdots ,x_n]_{\le 4d}\rightarrow \mathbb {R}\), in order to have a flat extension.

在本文中，我们改善了Carathéodory数的界线，特别是在代数变体且具有较小的间隙（并非所有单项式都存在）的情况下。我们提供了代数变体\（\ mathbb {R} ^ n \）和\（[0,1] ^ n \）的明确上下限。我们还以很小的差距来对待力矩问题。我们发现，对于每个\（\ varepsilon> 0 \）和\（d \ in \ mathbb {N} \），都有一个\（n \ in \ mathbb {N} \）这样我们就可以构造矩函数\ （L：\ mathbb {R} [x_1，\ cdots，x_n] _ {\ le d} \ rightarrow \ mathbb {R} \）至少需要\（（1- \ varepsilon）\ cdot \ left（{\ begin {matrix} n + d \\ n \ end {matrix}} \ right）\）原子\（l_ {x_i} \）。获得了汉克矩阵和平坦展开的结果和结果。我们发现存在矩函数\（L：\ mathbb {R} [x_1，\ cdots，x_n] _ {\ le 2d} \ rightarrow \ mathbb {R} \）需要扩展到最坏的情况4 d，\（\ tilde {L}：\ mathbb {R} [x_1，\ cdots，x_n] _ {\ le 4d} \ rightarrow \ mathbb {R} \），以便具有平坦的扩展名。