Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines

A real n-dimensional homogeneous polynomial f (x) of degree m and a real constant c define an algebraic hypersurface S whose points satisfy f (x) = c. The polynomial f can be represented by Axm where A is a real mth order n-dimensional supersymmetric tensor. In this paper, we define rank, base index and eigenvalues for the polynomial f , the hypersurface S and the tensor A. The rank is a nonnegative integer r less than or equal to n. When r is less than n, A is singular, f can be converted into a homogeneous polynomial with r variables by an orthogonal transformation, and S is a cylinder hypersurface whose base is r -dimensional. The eigenvalues of f , A and S always exist. The eigenvectors associated with the zero eigenvalue are either recession vectors or degeneracy vectors of positive degree, or their sums. When c 6= 0, the eigenvalues with the same sign as c and their eigenvectors correspond to the characterization points of S, while a degeneracy vector generates an asymptotic ray for the base of S or its conjugate hypersurface. The base index is a nonnegative integer d less than m. If d = k, then there are nonzero degeneracy vectors of degree k−1, but no nonzero degeneracy vectors of degree k. A linear combination of a degeneracy vector of degree k and a degeneracy vector of degree j is a degeneracy vector of degree k+ j−m if k+ j ≥ m. Based upon these properties, we classify such algebraic hypersurfaces in the nonsingular case into ten classes. c © 2006 Elsevier Ltd. All rights reserved.