Exponent-Incidence Constraints for Tensor Eigenvectors of Multi-Hypergraphs

We study support-determined constraints for eigenvectors of nonnegative symmetric tensors whose support may contain repeated indices. Such tensors are naturally encoded by uniform multi-hypergraphs, where each multiedge is represented by an exponent vector α∈N0n with |α|=k. Replacing the ordinary vertex-edge incidence matrix by the exponent-incidence matrix, we show that every nonzero-eigenvalue H-eigenvector satisfies linear incidence constraints in the transformed coordinates yi=xik. These constraints are invariant under positive scalar edge weights and reduce to the usual support-incidence constraints for ordinary squarefree hypergraphs. When the exponent-incidence matrix has a nontrivial left kernel, these relations provide nonzero support-level constraints in the transformed coordinates. We also describe the positive branch of the resulting constraint variety and prove a positive-weight realization criterion: a positive vector x can be realized as an eigenvector of some strictly positive edge-weighting of a fixed multi-hypergraph if and only if x[k] lies in the strictly positive coefficient cone generated by the exponent-incidence columns. Thus, the exponent-incidence constraint variety provides necessary algebraic constraints in general, while the strictly positive coefficient cone gives the exact positive-weight realization region on the positive branch.