Fast computation of linear finite-dimensional operators over arbitrary rings

Abstract

Invention of the FFT-algorithm [4] suggests that some linear transforms over a ring $\mathbb{K}$,

$$T:\eta = {\rm A}\xi ,\xi \varepsilon \mathbb{K}^n ,\eta \varepsilon \mathbb{K}^m ,A = (a_{ij} ),a_{ij} \varepsilon \mathbb{K},l \leqslant i \leqslant m,l \leqslant j \leqslant n,$$

((*))

could be computed faster than by the definition algorithm

$$\eta _i = a_{il} \cdot \xi _l + \ldots + a_{in} \cdot \xi _n ,l \leqslant i \leqslant m.$$

The Main Problem : Let $\mathbb{L}$be an extension of $\mathbb{K},\mathbb{K} \subseteq \mathbb{L}$. Find a shortest linear circuit over $\mathbb{L}$computing T (*).

We construct an algorithm which reduces this problem to the following one : for a given system of polynomial equations of the form

$$f_k (u_l , \ldots ,u_q ) = \alpha _k ,f_k \varepsilon \mathbb{Z}[u_l , \ldots ,u_q ],\alpha _k \varepsilon \mathbb{K},l \leqslant k \leqslant p,$$

((**))

either find its solution in $\mathbb{L}$or prove that such a solution does not exist.

Therefore, when the problem (**) is algorithmically solvable, so is our main problem, — as, e.g., when $\mathbb{K} = \mathbb{Q},\mathbb{L} = \mathbb{R}$or $\mathbb{K} = \mathbb{L}$=GF(q); in both cases we give explicit (exponential) upper estimates on the complexity of the full algorithms solving the corresponding main problem. We consider also implications of actual or plausible algorithmical unsolvability of the problem (**) for our main problem.