Geometry-Driven Conditioning of Multivariate Vandermonde Matrices in High-Degree Regimes

We study multivariate monomial Vandermonde matrices $V_N(Z)$ with arbitrary distinct nodes $Z=\{z_1,\dots ,z_s\}\subset B_2^n$ in the high-degree regime $N\ge s-1$. Introducing a projection-based geometric statistic – the \emph{max-min projection separation} $\rho (Z,j)$ and its minimum $\kappa (Z)={\min }_j\rho (Z,j)$ – we construct Lagrange polynomials $Q_j\in {\mathcal{P}}_N^n$ with explicit coefficient bounds $$\mathrm{\parallel }{Q}_j{\mathrm{\parallel }}_{\mathrm{\infty }}\lesssim s(\frac{4n}{\rho (Z,j)}{)}^{s-1}\mathrm{.}$$ These polynomials yield quantitative distance-to-span estimates for the rows of $V_N(Z)$ and, as consequences, $${\sigma }_{\min }(V_N(Z))\gtrsim \frac{\kappa (Z{)}^{s-1}}{(4n{)}^{s-1}s\sqrt{s\nu (n,N)}},\nu (n,N)=\left(\genfrac{}{}{0px}{}{N+n}{N}\right),$$ and an explicit right inverse $V_N(Z{)}^{+}$ with operator-norm control $$\mathrm{\parallel }{V}_N(Z{)}^{+}\mathrm{\parallel }\lesssim s^{3\mathrm{/}2}\sqrt{\nu (n,N)}(\frac{4n}{\kappa (Z)}{)}^{s-1}\mathrm{.}$$ Our estimates are dimension-explicit and expressed directly in terms of the local geometry parameter $\kappa (Z)$; they apply to \emph{every} distinct node set $Z\subset B_2^n$ without any \emph{a priori} separation assumptions. In particular, $V_N(Z)$ has full row rank whenever $N\ge s-1$. The results complement the Fourier-type theory (on the complex unit circle/torus), where lower bounds for ${\sigma }_{\min }$ hinge on uniform separation or cluster structure; here stability is quantified instead via high polynomial degree and the projection geometry of $Z$.