Algorithms for the universal decomposition algebra

Romain Lebreton (1) , Éric Schost (2)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 ORCCA

Let k be a field and let f be a polynomial of degree n in k [T]. The symmetric relations are the polynomials in k [X1, ..., Xn] that vanish on all permutations of the roots of f in the algebraic closure of k. These relations form an ideal Is; the universal decomposition algebra is the quotient algebra A := k [X1, ..., Xn]/Is. We show how to obtain efficient algorithms to compute in A. We use a univariate representation of A, i.e. an explicit 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.

Keywords: Algorithms; complexity; universal decomposition algebra; lagrange resolvent; cauchy modules; divided differences; invariants

Domains: Symbolic Computation [cs.SC]

Romain Lebreton, Éric Schost. Algorithms for the universal decomposition algebra. 2012. ⟨hal-00669806⟩