On the Annihilator Ideal of an Inverse Form

TL;DR

This paper studies the annihilator ideal of inverse forms over a field, extending previous work by constructing generators, algorithms, and bases for these ideals, with applications to finite sequences and LFSR synthesis.

Contribution

It introduces a new approach to analyze inverse forms as modules, providing explicit generators, algorithms, and connections to the Berlekamp-Massey algorithm for sequence analysis.

Findings

Constructed generators for the annihilator ideal with specific divisibility properties.

Developed an efficient algorithm for computing the Gr"obner basis of the ideal.

Linked the algebraic structures to sequence analysis and LFSR synthesis.

Abstract

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.