1 results
ON THE BINARY SEQUENCE
$(1,1,0,1,0^3,1,0^7,1,0^{15},\ldots )$
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 23 October 2024, pp. 1-12
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Let
$\mathbb {F}$ be a field and 
$(s_0,\ldots ,s_{n-1})$ be a finite sequence of elements of 
$\mathbb {F}$. In an earlier paper [G. H. Norton, ‘On the annihilator ideal of an inverse form’, J. Appl. Algebra Engrg. Comm. Comput. 28 (2017), 31–78], we used the 
$\mathbb {F}[x,z]$ submodule 
$\mathbb {F}[x^{-1},z^{-1}]$ of Macaulay’s inverse system 
$\mathbb {F}[[x^{-1},z^{-1}]]$ (where z is our homogenising variable) to construct generating forms for the (homogeneous) annihilator ideal of 
$(s_0,\ldots ,s_{n-1})$. We also gave an 
$\mathcal {O}(n^2)$ algorithm to compute a special pair of generating forms of such an annihilator ideal. Here we apply this approach to the sequence r of the title. We obtain special forms generating the annihilator ideal for 
$(r_0,\ldots ,r_{n-1})$ without polynomial multiplication or division, so that the algorithm becomes linear. In particular, we obtain its linear complexities. We also give additional applications of this approach.
