  Let $K$ be a field. We simplify and extend work of Althaler \& D\"ur on
finite sequences over $K$ by regarding $K[x^{-1},z^{-1}]$ as a $K[x,z]$ module,
and studying forms in $K[x^{-1},z^{-1}]$ from first principles. Then we apply
our results to finite sequences.
  First we define the annihilator ideal $I_F$ of a non-zero form $F\in
K[x^{-1},z^{-1}]$, a homogeneous ideal. We inductively construct an ordered
pair ($f_1$\,,\,$f_2$) of forms which generate $I_F$\,; our generators are
special in that $z$ does not divide the leading grlex monomial of $f_1$ but $z$
divides $f_2$\,, and the sum of their total degrees is always $2-|F|$, where
$|F|$ is the total degree of $F$. We show that $f_1,f_2$ is a maximal regular
sequence for $I_F$, so that the height of $I_F$ is 2. The corresponding
algorithm is $\sim |F|^2/2$.
  The row vector obtained by accumulating intermediate forms of the
construction gives a minimal grlex Gr\"obner basis for $I_F$ for no extra
computational cost other than storage and apply this to determining $\dim_K
(K[x,z] /I_F)$\,. We show that either the form vector is reduced or a monomial
of $f_1$ can be reduced by $f_2$\,. This enables us to efficiently construct
the unique reduced Gr\"obner basis for $I_F$ from the vector extension of our
algorithm.
  Then we specialise to the inverse form of a finite sequence, obtaining
generator forms for its annihilator ideal and a corresponding algorithm which
does not use the last 'length change' of Massey. We compute the intersection of
two annihilator ideals using syzygies in $K[x,z]^5$. This improves a result of
Althaler \& D\"ur. Finally, dehomogenisation induces a one-to-one
correspondence ($f_1$\,,$f_2$) $\mapsto$ (minimal polynomial, auxiliary
polynomial), the output of the author's variant of the Berlekamp-Massey
algorithm. So we can also solve the LFSR synthesis problem via the
corresponding algorithm for sequences.
