Generalised symmetries of polynomials in algebraic complexity

Abstract

Suppose a polynomial P(x), (where x is the column matrix of the indeterminates x1.~.x), has a symmetry analogous to those of the de~ermiRant, whereby taking a certain linear combination of the The purpose of the paper is to define the set of symmetries of a polynomial, explore its structure, and identify the computationally useful members; a method of computing the latter symmetries is presented. It is shown how the set of symmetries determines whether or not Gaussian style elimination or transformation style algorithms can aid computation. To this end a robust notion of the "dimen-sion" of a polynomial is defined, yielding a tech~ nique for proving negative results in complexity. Let ~'¥.. and z be n x 1 column matrices. z is the wrappeg convolution of ~ and ¥.. iff '1j z. = LX. Y.. d • z is the Hademard J i=l ~ ~+J mo n product (or pairwise product) of ~ and ¥.."iff '1j z. = x. • y j • An efficient technique for eval-J J uating a wrapped convolution [1 p.254] relies upon transforming the convolution into a Hademard product by means of the discrete "Fburier transform. The question "can the permanent be transformed analogously in a way that may assist faster computation?" is considered, and answered in part. In order to construct such a scheme whereby P can be evaluated at any point ~, it must be possible for the symmetry (T, t) to depend upon x, in order to introduce zeros into Tx + t. (In practice several successive transformations may be made, introducing successively more zeros whilst preserving those previously present. Such a scheme constitutes a Gaussian elimination style algorithm for evaluating P) • In order for this to be possible, it is necessary that some of the symmetries of P form a continuum: these continuous symmetries include all of the computationally useful symmetries of P. variables before evaluating P only alters the result by a constant factor plus a constant additive term, (where T is an n x n matrix of constants, t is an n X 1 matrix of constants and k,k' are constants) • Then P could be computed at ~ by computing P at Ta + t, multiplying by k and adding k'. If Ta + t has more components equal to zero than a then there may be some computational advantage in this scheme as compared to evaluating …