Square-free decomposition in finite characteristic

Abstract

In ([GT]) has been addressed the problem of the computation of the square-free decomposition for univariate polynomials with coefficients in arbitrary fields. The complete square-free decomposition can be computed over arbitrary fields of finite characteristic solely assuming that the field satisfies the Condition P of Seidenberg ([Se]), which has been proven equivalent to the ability computing such decompositions (see also [MRR]). If we assume that the field is only an effective field (i.e. of a field K where there are constructive procedures for performing rational operations in K and for deciding whether or not two elements in K are equal), it is possible to obtain a weaker decomposition into powers of relatively prime factors, not necessarily square-free, but such that within each factor the roots have constant multiplicity. Although this is a partial decomposition, much useful information can be gathered from this result. As an application we present an algorithm to compute the Jordan form of a matrix over an arbitrary effective field. In particular we show how to handle problems of inseparability while splitting invariant factors and constructing symbolic Jordan form. The computation of normal forms of a matrix, in particular of the Jordan form, is a very important task and has many useful applications, so it has been widely studied for many years and many efficient algorithms, sequential and parallel ([O], [L], [Gi1], [Gi2], [Ol], [KKS], [RV]), are already available for its computation. There are already algorithms which compute the Jordan form of a matrix over general fields ([GD], [RV]), but they are based on dynamic evaluation ([D5]) and we want to avoid the use of such a scheme, that requires a special computational environment. Storjohann ([St]) has given a new algorithm for computing the rational canonical form which has a deterministic complexity of O ( n 3 ) but he does not compute the transition matrix with the same complexity. Steel's ([S]) algorithm for computing generalized Jordan form has a complexity O ( n 4 ) but requires factoring polynomials into irreducibles. Kaltofen et. al. ([KKS]) give fast parallel algorithms for canonical forms and make the observation that one could compute a symbolic Jordan form from a rational canonical form by splitting the invariant factors using gcd's and square-free decompositions. They require the computation of complete square-free decompositions and thus also require that K be a perfect field with the ability to compute p th roots. They also don't compute the transition matrix. Ozello ([O]) presents an algorithm for computing the rational canonical form which is deterministic with complexity O ( n 4 ), and leaves the question of faster probabilitic approaches for future work. Giesbrecht ([Gi2]) gives a probabilistic algorithm whose complexity is essentially the same as matrix multiplication but requires choosing n "good" random vectors simultaneously thus giving only a probability of 1/4 of making a successful choice. Our aim is to obtain a general sequential algorithm, of a complexity comparable with most of the existing algorithms, that works in the widest possible setting, without requiring particular computing resources and hence of easy and straightforward implementation. Because of our hypothesis, in general, our algorithm will produce a symbolic Jordan form ([K], [RV]), but the main difference with the other available algorithms based on dynamic evaluation is that our algorithm is a rational algorithm, since all the computations take place in the given field, except for the output and eventually the computation of the inverse of the transition matrix. To obtain all the information on the symbolic roots of the characteristic polynomial (multiplicities and recognition) we, at first, transform the given matrix A into a pseudo-rational form, i.e. a block diagonal matrix, similar to A, with companion matrices on the diagonal without requiring any kind of divisibility of the associated polynomials. Then we refine the factorization of the characteristic polynomial, given by the polynomials whose companion matrices are on the diagonal of the pseudo-rational form, using partial square-free decomposition and gcd computations, so that we can identify the same roots in different blocks and also we reduce, as much as possible without factorization, the degree of the defining polynomials for the eigenvalues. The pseudo-rational form is computed with a probabilistic algorithm of complexity O ( n 3 ) such that each independent random choice is verifiable with probability better than 1 - 1/ n of success. We derive this probabilistic algorithm from one for the computation of the rational form, which has a complexity of O ( n 4 ), and is obtained via a straightforward analysis of the properties of the minimal polynomial that leads to a natural way to construct invariant subspaces.