Primary decomposition of zero-dimensional ideals over finite fields

Abstract

Abstract.A new algorithm is presented for computing primary decomposition of zero-dimensional ideals over finite fields. Like Berlekamp's algorithm for univariate polynomials, the new method is based on the invariant subspace of the Frobenius map acting on the quotient algebra. The dimension of the invariant subspace equals the number of primary components, and a basis of the invariant subspace yields a complete decomposition. Unlike previous approaches for decomposing multivariate polynomial systems, the new method does not need primality testing nor any generic projection, instead it reduces the general decomposition problem directly to root finding of univariate polynomials over the ground field. Also, it is shown how Gröbner basis structure can be used to get partial primary decomposition without any root finding.

An efficient algorithm for factoring polynomials over algebraic extension field

“…That is we can factor the polynomials in finite field first, and lift the factorization to characteristic 0 afterwards. We also notice that Gao gives an efficient algorithm for computing the primary decomposition over finite fields (Gao et al, 2009), which may help to improve the new algorithm in finite field and hence benefits for our future work.…”

Computing and using minimal polynomials

“…The following algorithm makes a good use of Frob q (P/I), the Frobenius space of P/I. Inspired by Gao et al (2008), the detailed theoretical and computational aspects related to this concept are described in Kreuzer and Robbiano (2016), Section 5.2. Algorithm 4.29.…”

An Alternative Method for Primary Decomposition of Zero-dimensional Ideals over Finite Fields

“…Gao, Wan and Wang in [7] present an interesting approach to compute primary decomposition of zero-dimensional ideals over finite fields. The method is based on the invariant subspace of the Frobenius map acting on the quotient algebra k[x]/I.…”

“…In the method in [7], if one chooses an element of the basis of the invariant subspace which is separable for I, then all the primary components can be computed at once. Otherwise, the further decomposition is necessary even though the probability of separable element in the invariant subspace is not low in most cases, see Proposition 3.2 in [7] for details.…”

“…In the method in [7], if one chooses an element of the basis of the invariant subspace which is separable for I, then all the primary components can be computed at once. Otherwise, the further decomposition is necessary even though the probability of separable element in the invariant subspace is not low in most cases, see Proposition 3.2 in [7] for details. In this paper, we aim to find an approach to decompose the above invariant subspace into a direct product of several one-dimensional k−algebras in Proposition 1 in Section 3.…”