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On the Annihilator Ideal of an Inverse Form

by   Graham H. Norton, et al.
The University of Queensland

Let K be a field. We simplify and extend work of Althaler & Dür 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∈ 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 ∼ |F|^2/2. The row vector obtained by accumulating intermediate forms of the construction gives a minimal grlex Gröbner basis for I_F for no extra computational cost other than storage and apply this to determining _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 Gröbner 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. We compute the intersection of two annihilator ideals using syzygies in K[x,z]^5. This improves a result of Althaler & Dür. Finally, dehomogenisation induces a one-to-one correspondence (f_1 ,f_2) (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.


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