Parameter Counts and Circuit Constraint Spaces in Spectral Hypergraph Theory

This paper studies eigenvector constraints for adjacency hypermatrices of uniform hypergraphs using matroid circuits. Building on Pearson’s circuit relations, we define the circuit constraint space L(H) as the span of all Pearson relation vectors associated to circuits of the transversal matroid of the Levi graph. This yields a basis-independent parameter count π(H):=|V(H)|−dim(L(H)), which bounds the degrees of freedom for transformed eigenvectors associated with nonzero eigenvalues. We compare L(H) with basis-dependent spaces LB(ΛB), obtained by choosing one Pearson relation from each fundamental circuit relative to a matroid basis. Under a triangular independence condition and constraint-efficiency, we prove π(H)=rank(M(H)). We verify this equality for loose paths, hyperstars, and sunflowers. We also develop a constraint ideal interpretation, study symmetry via automorphism actions, and use the resulting constraints to construct structured initializations for shifted higher-order power methods. The numerical examples illustrate how these structured initializations can influence observed iteration dynamics, without establishing a general convergence or acceleration theorem.