Think of your matrix as the evaluation matrix of the bivariate polynomial space Π = span{x^b y^a : 0 ≤ a ≤ m−1, 0 ≤ b ≤ m−1}, which has dimension m^2. Your matrix V has one column per point (x_i, y_i) and one row per monomial in Π, so V ∈ R^{m^2 × (m^2+1)}.

1. Dimension of the kernel

• For “generic” points (x_i, y_i) (i.e., away from an algebraic exceptional set), rank(V) = m^2, hence dim ker(V) = (m^2+1) − m^2 = 1.
• Rank(V) < m^2 (so ker dimension > 1) if and only if every m^2 × m^2 minor vanishes. Equivalently, for every subset of m^2 of the points there is a nonzero p ∈ Π that vanishes on that subset. This requires special configurations (e.g., many points lying on the zero set of some p ∈ Π).
• So no extra condition like x_i^a y_i^b ≠ x_j^c y_j^d is needed in general; full rank is the generic situation.

2. An explicit spanning vector of ker(V) There are two standard, completely general descriptions.

• Cofactor/minor formula (works for any k × (k+1) matrix of full row rank): Let V_i be the m^2 × m^2 submatrix obtained by deleting the i-th column of V. Then a nonzero vector in ker(V) is c = (c_1, …, c_n) with c_i = (−1)^{i−1} det(V_i). This is the direct analogue of the univariate Vandermonde case. In 1D the minors factor into products of differences, yielding the familiar closed form; in several variables there is generally no such simple factorization unless the nodes have special structure.

• Lagrange-formula via unisolvent subset: Choose any index s such that the square submatrix formed by the other m^2 columns is invertible (this exists generically). On those m^2 points, there are unique Lagrange fundamental polynomials L_i(x, y) ∈ Π, i ≠ s, satisfying L_i(x_j, y_j) = δ_{ij} for j ≠ s. Then the s-th column is a linear combination of the others: [values at (x_s, y_s)] = Σ_{i≠s} L_i(x_s, y_s) [values at (x_i, y_i)]. Hence a kernel vector is given by c_s = 1, and c_i = −L_i(x_s, y_s) for i ≠ s. This is the natural multivariate analogue of the univariate barycentric description (but now in terms of multivariate Lagrange polynomials, not simple products).

3. When do simpler closed forms exist?

• In general, unlike the 1D Vandermonde, there is no universal product-of-differences formula. Such simplifications occur only in special configurations (e.g., when the points are parameterized by a single variable and the sampled monomials reduce effectively to a univariate basis, or on certain tensor-product structures with appropriately chosen spaces).
• For arbitrary distinct points, the cofactor formula (or the Lagrange formula relative to an invertible m^2-point subset) is the natural “analytic” description.

Summary:

• Generically, ker(V) is 1-dimensional.
• A spanning vector is given by signed m^2 × m^2 minors c_i = (−1)^{i−1} det(V_i), or equivalently via multivariate Lagrange polynomials relative to any invertible m^2-column subset.
• No additional constraints like x_i^a y_i^b ≠ x_j^c y_j^d are needed beyond avoiding the (measure-zero) configurations where all m^2 × m^2 minors vanish.