Square-free decomposition in finite characteristic: an application to Jordon Form computation

In ([GT]) has been addressed the problem of the computation ofthe square-free decomposition for univariate polynomials withcoefficients in arbitrary fields. The complete square-freedecomposition can be computed over arbitrary fields of finitecharacteristic solely assuming that the field satisfies theCondition P of Seidenberg ([Se]), which has been provenequivalent 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 proceduresfor performing rational operations in K and for decidingwhether or not two elements in K are equal), it is possibleto obtain a weaker decomposition into powers of relatively primefactors, not necessarily square-free, but such that within eachfactor the roots have constant multiplicity. Although this is apartial decomposition, much useful information can be gathered fromthis result. As an application we present an algorithm to computethe Jordan form of a matrix over an arbitrary effective field. Inparticular we show how to handle problems of inseparability whilesplitting invariant factors and constructing symbolic Jordanform. The computation of normal forms of a matrix, in particular ofthe Jordan form, is a very important task and has many usefulapplications, so it has been widely studied for many years and manyefficient algorithms, sequential and parallel ([O], [L], [Gi1],[Gi2], [Ol], [KKS], [RV]), are already available for itscomputation. There are already algorithms which compute the Jordanform of a matrix over general fields ([GD], [RV]), but they arebased on dynamic evaluation ([D5]) and we want to avoid the use ofsuch a scheme, that requires a special computational environment.Storjohann ([St]) has given a new algorithm for computing therational canonical form which has a deterministic complexity ofO(n3) but he does not compute thetransition matrix with the same complexity. Steel's ([S]) algorithmfor computing generalized Jordan form has a complexityO(n4) but requires factoring polynomialsinto irreducibles. Kaltofen et. al. ([KKS]) give fast parallelalgorithms for canonical forms and make the observation that onecould compute a symbolic Jordan form from a rational canonical formby splitting the invariant factors using gcd's and square-freedecompositions. They require the computation of completesquare-free decompositions and thus also require that K be aperfect field with the ability to compute pthroots. They also don't compute the transition matrix. Ozello ([O])presents an algorithm for computing the rational canonical formwhich is deterministic with complexityO(n4), and leaves the question of fasterprobabilitic approaches for future work. Giesbrecht ([Gi2]) gives aprobabilistic algorithm whose complexity is essentially the same asmatrix multiplication but requires choosing n "good" randomvectors simultaneously thus giving only a probability of 1/4 ofmaking a successful choice. Our aim is to obtain a general sequential algorithm, of acomplexity comparable with most of the existing algorithms, thatworks in the widest possible setting, without requiring particularcomputing resources and hence of easy and straightforwardimplementation. Because of our hypothesis, in general, ouralgorithm will produce a symbolic Jordan form ([K], [RV]),but the main difference with the other available algorithms basedon dynamic evaluation is that our algorithm is a rationalalgorithm, since all the computations take place in the givenfield, except for the output and eventually the computation of theinverse of the transition matrix. To obtain all the information onthe symbolic roots of the characteristic polynomial (multiplicitiesand recognition) we, at first, transform the given matrix Ainto a pseudo-rational form, i.e. a block diagonal matrix,similar to A, with companion matrices on the diagonalwithout requiring any kind of divisibility of the associatedpolynomials. Then we refine the factorization of the characteristicpolynomial, given by the polynomials whose companion matrices areon the diagonal of the pseudo-rational form, using partialsquare-free decomposition and gcd computations, so that we canidentify the same roots in different blocks and also we reduce, asmuch as possible without factorization, the degree of the definingpolynomials for the eigenvalues. The pseudo-rational form is computed with a probabilisticalgorithm of complexity O(n3) such thateach independent random choice is verifiable with probabilitybetter than 1 - 1/n of success. We derive this probabilisticalgorithm from one for the computation of the rational form, whichhas a complexity of O(n4), and is obtainedvia a straightforward analysis of the properties of the minimalpolynomial that leads to a natural way to construct invariantsubspaces.