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 $$\|Q_j\|_\infty \lesssim s\Bigl(\frac{4n}{\rho(Z,j)}\Bigr)^{s-1}.$$ These polynomials yield quantitative distance-to-span estimates for the rows of $V_N(Z)$ and, as consequences, $$\sigma_{\min}(V_N(Z)) \gtrsim \frac{\kappa(Z)^{s-1}}{(4n)^{s-1} s\sqrt{s \nu(n,N)}}, \quad \nu(n,N)={N+n\choose N},$$ and an explicit right inverse $V_N(Z)^+$ with operator-norm control $$\|V_N(Z)^+\| \lesssim s^{3/2}\sqrt{\nu(n,N)}\Bigl(\frac{4n}{\kappa(Z)}\Bigr)^{s-1}.$$ 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 $Z$.