On Determinantal Varieties of Hankel Matrices

Let be a class of Hankel matrices whose entries, depending on a given matrix , are linear forms in variables with coefficients in a finite field . For every matrix in , it is shown that the varieties specified by the leading minors of orders from 1 to have the same number of points in . Further properties are derived, which show that sets of varieties, tied to a given Hankel matrix, resemble a set of hyperplanes as regards the number of points of their intersections. 1. Introduction The representation of hypersurfaces of small degree as determinants is a classical subject. For instance, Hesse [1] discussed the representation of the plane quartic by symmetric determinants, and many different problems have been tackled over the years; see, for example, [2, 3]. An important question, when hypersurfaces are defined over finite fields, is the computation of the number of points. In general this is very difficult, for example, [4], and most frequently only bounds are given. This paper considers hypersurfaces over finite fields, which are defined by determinants of Hankel matrices whose entries are linear forms in the variables. These Hankel matrices are encountered in the proof of certain properties of finite state automata whose state change is governed by tridiagonal matrices [5, 6]. They also occur in the study of some decoding algorithms for error-correcting codes [7, 8]. It is remarkable that, for these determinantal varieties, the exact number of points can in many instances be explicitly found, in terms of the size of the field and the number of variables. Let be an irreducible polynomial of degree over with root , which is thus an eigenvalue of the companion matrix which is assumed to have the coefficients of in the last column, all s in the first subdiagonal, and the remaining entries are s [9]. The definition of Hankel matrices that we are dealing with uses the Krylov matrices where is a row vector of independent variables and is a column vector of independent variables. Every Krylov matrix is nonsingular unless and are all-zero vectors, as will be proved later. Definition 1. The class consists of matrices defined as These are Hankel matrices, because the entries are clearly the same whenever the index sum is constant. When the vector is a fixed element of , the corresponding subclass of is denoted by . Given a polynomial in the ring , the variety is defined as the set of points in the affine space that annihilate ; that is, More generally, given polynomials the variety is the set of solutions of the system Note that is the intersection and is the