Algorithms for the universal decomposition algebra
Romain Lebreton, Éric Schost · 2012
Let k be a field and let f ∈ k [T] be a polynomial of degree n. The universal decomposition algebra A is the quotient of k [X1,...,Xn] by the ideal of symmetric relations (those polynomials that vanish on all permutations of the roots of f). We show how to obtain efficient algorithms to compute in A. We use a univariate representation of A, i.e. an isomorphism of the form A k[T]/Q(T), since in this representation, arithmetic operations in A are known to be quasi-optimal. We give details for two related algorithms, to find the isomorphism above, and to compute the characteristic polynomial of any element of A.