The dimension of eigenvariety of nonnegative tensors associated with spectral radius

For a nonnegative weakly irreducible tensor, its spectral radius is an eigenvalue corresponding to unique positive eigenvector up scalar, called Perron vector. But including the vector, it may have more than one radius. The projective eigenvariety associated with set of eigenvectors considered in complex space. We prove that dimension above zero, i.e. there are finitely many scalar. characterize combinatorially symmetric tensor for which greater zero. Finally we apply those results adjacency tensors uniform hypergraphs.