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

Abstract

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 A x m where A is a real m th 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 ⁄ = 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.