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M r ogun to congruent are ilbe will p ≤ ai numbers -adic = p 11 α N O DQ P 1 ,tecom- the ], ( sln as long as p P ≤ N i k then ) =1 The . α χ 2 e.g. α M ing ≤ i - . , polynomials is often a little bit less thoroughly studied: some and χ the characteristic polynomial map, χM ∈ K[X] for the refer to fast algorithms (using division), while others apply characteristic polynomial of M and dχM for the differential division-free algorithms. of χ at M, as a linear map from Mn(K) to the space of In [4], the authors have begun the application of the the- polynomials of degree less than n. Wefixan ω ∈ R such that ory of differential precision of [3] to the stable computation the multiplication of two matrices over a ring is in O(nω ) of characteristic polynomials. They have obtained a way to operations in the ring. Currently, the smallest known ω is express the optimal precision on the characteristic polyno- less than 2.3728639 thanks to [13]. We will denote by In mial, but have not given practical algorithms to attain this the identity matrix of rank n in Mn(K). When there is no optimal precision. ambiguity, we will drop this In for scalar matrices, e.g. for λ ∈ K and M ∈ Mn(K), λ − M denotes λIn − M. Finally, The contribution of this paper. we write σ1(M),...,σn(M) for the elementary divisors of Thanks to the application the framework of differential pre- M, sorted in increasing order of valuation. cision of [3] in [4], we know that the precision of the char- acteristic polynomial χM of a matrix M ∈ Mn(Qp) is de- 2. THEORETICAL STUDY termined by the comatrix Com(X − M). In this article, we provide: 2.1 The theory of p-adic precision 1. Proposition 2.7: a factorization of Com(X − M) as a We recall some of the definitions and results of [3] as a product of two rank-1 matrices (when M has a cyclic foundation for our discussion of the precision for the charac- vector), computable in O˜(nω) operations by Theorem teristic polynomial of a matrix. We will be concerned with 4.1 two K-manifolds in what follows: the space Mn(K) of n × n matrices with entries in K and the space Kn[X] of monic 2. Corollary 2.4: a simple, O˜(nω) criterion to decide degree n polynomials over K. Given a matrix M ∈ Mn(K), whether χM is defined at precision higher than the the most general kind of precision structure we may attach precision of M (when M ∈ Mn(Zp)). to M is a lattice H in the tangent space at M. However, rep- 3. Theorem 3.11: a O˜(n3) algorithm with operations in resenting an arbitrary lattice requires n2 basis vectors, each 2 Zp to compute the optimal precision on each coefficient with n entries. We therefore frequently work with certain of χM (when M is given with uniform precision on its classes of lattices, either jagged lattices where we specify a entries). precision for each matrix entry or flat lattices where every N ω entry is known to a fixed precision O(p ). Similarly, pre- 4. Proposition 5.6: a O˜(n ) algorithm to compute the cision for monic polynomials can be specified by giving a optimal precision on each eigenvalue of M lattice in the tangent space at f(X) ∈ Kn[X], or restricted Organization of the article. to jagged or flat precision in the interest of simplicity. Let χ : Mn(K) → Kn[X] be the characteristic polynomial In Section 2, we review the differential theory of precision map. Our analysis of the precision behavior of χ rests upon developed in [3] and apply it to the specific case of the char- the computation of its derivative dχ, using [3, Lem. 3.4]. For acteristic polynomial, giving conditions under which the dif- a matrix M ∈ Mn(K), we identify the tangent space V at ferential will be surjective (and thus provide a good measure M with Mn(K) itself, and the tangent space W at χM with of precision). We also give a condition based on reduction the space K
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