Estimation of a quasi-Vandermonde determinant

Lemma 1. If $0\leq k_{1}

$\det\left(\left[\begin{array}{cccc} \lambda_{1}^{k_{1}} & \lambda_{1}^{k_{2}} & \cdots & \lambda_{1}^{k_{n}}\\ \lambda_{2}^{k_{1}} & \lambda_{2}^{k_{2}} & \cdots & \lambda_{2}^{k_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ \lambda_{n}^{k_{1}} & \lambda_{n}^{k_{2}} & \cdots & \lambda_{n}^{k_{n}} \end{array}\right]\right)$

can be estimated as

$h(\lambda_{\overline{1,n}},n_{\lambda})\prod_{1\leq j_{1}0$

where $h(\lambda_{\overline{1,n}},n_{\lambda})$ is a homogeneous polynomials with the order $n_{\lambda}$ on the $n$ -variables $\lambda_{\overline{1,n}}=\{\lambda_{1},\lambda_{2},\cdots,\lambda_{n}\}$ , and can be represented as

$h(\lambda_{\overline{1,n}},n_{\lambda})=\sum_{\sum_{j=1}^{n}s_{j}=n_{\lambda}}g_{s_{\overline{1,n}}}\prod_{j=1}^{n}\lambda_{j}^{s_{j}}$

                              $n_{\lambda}=\sum_{j=1}^{n}\left(k_{j}-j+1\right)$

where $s_{\overline{1,n}}=\{s_{1},s_{2},\cdots,s_{n}\}$ , all $g_{s_{\overline{i,n}}}\geq0$ and there exist some $g_{s_{\overline{i,n}}}>0$ .

The above result was given and proven, in Apr. 2017, and then is used in the volume computing of a class of special polyhedrons.