Finite Linear Recurring Sequences and Homogeneous Ideals

The linear recurrence relations satisfied by finitely many sequences of finite length over a ground field are described by homogeneous ideals in the polynomial ring in two variables by using Macaulay's theory of inverse systems. The class of these ideals is shown to be precisely the class of homogeneous primary ideals the associated prime of which is the irrelevant maximal ideal. In the case of a single sequence, the classical Berlekamp-Massey algorithm for linear feedback shift register synthesis can be applied to obtain a minimal Gröbner basis of the ideal. The case of multiple sequences is reduced to the case of single sequences by ideal intersection, and the set of all linear recurrence relations of minimal order for the given sequences is generated by the low degree polynomials of the Gröbner basis.