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

Abstract

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} $\ddot{I}(Z,j)$ and its minimum $\hat{I}º(Z)={\min }_j\ddot{I}(Z,j)$ -- we construct Lagrange polynomials $Q_j\in {\mathcal{P}}_N^n$ with explicit coefficient bounds $$\Vert Q_j{\Vert }_{\infty }\lesssim s(\frac{4n}{\ddot{I}(Z,j)}{)}^{s-1}.$$ These polynomials yield quantitative distance-to-span estimates for the rows of $V_N(Z)$ and, as consequences, $$\ddot{I}{ƒ}_{\min }(V_N(Z))\gtrsim \frac{\hat{I}º(Z{)}^{s-1}}{(4n{)}^{s-1}s\sqrt{s\hat{I}½(n,N)}},\hat{I}½(n,N)=\left(\genfrac{}{}{0ex}{}{N+n}{N}\right),$$ and an explicit right inverse $V_N(Z{)}^{+}$ with operator-norm control $$\Vert V_N(Z{)}^{+}\Vert \lesssim s^{3/2}\sqrt{\hat{I}½(n,N)}(\frac{4n}{\hat{I}º(Z)}{)}^{s-1}.$$ Our estimates are dimension-explicit and expressed directly in terms of the local geometry parameter $\hat{I}º(Z)$; 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 $\ddot{I}{ƒ}_{\min }$ hinge on uniform separation or cluster structure; here stability is quantified instead via high polynomial degree and the projection geometry of $Z$.