p-Adic Stability In Linear Algebra

Using the differential precision methods developed previously by the same authors, we study the p-adic stability of standard operations on matrices and vector spaces. We demonstrate that lattice-based methods surpass naive methods in many applications, such as matrix multiplication and sums and intersections of subspaces. We also analyze determinants , characteristic polynomials and LU factorization using these differential methods. We supplement our observations with numerical experiments.

💡 Research Summary

The paper investigates the problem of p‑adic precision loss in basic linear‑algebraic operations and proposes a lattice‑based framework that dramatically improves stability compared with the traditional coordinate‑wise tracking used in most computer‑algebra systems. Building on the authors’ earlier work on differential precision, the authors first recall the key theoretical tool: if a map f between finite‑dimensional K‑vector spaces is differentiable at a point v₀ and its differential f′(v₀) is surjective, then for any sufficiently small lattice H around the origin the image of the perturbed point v₀+H is exactly f(v₀)+f′(v₀)(H). This result (Proposition 2.1) guarantees that precision can be propagated with no loss by applying the differential to the input lattice. When f is an integral polynomial, a concrete bound on the admissible radius is given (Proposition 2.2).

Armed with this machinery, the authors analyze several standard operations:

1. Matrix multiplication – For A∈M_{r,s}(K) and B∈M_{s,t}(K) the differential is (dA,dB)↦A·dB+dA·B. Applying the differential to the standard lattice of integer‑valued matrices yields a sub‑lattice described by the Smith valuations a_i of A and b_j of B: the (i,j) entry of the product gains at least min(a_i,b_j) extra p‑adic digits. This “diffuse precision” is invisible to coordinate‑wise methods, which treat each entry independently and therefore accumulate loss linearly with the number of multiplications. Experiments (Algorithm 1) show that the lattice method’s loss grows only logarithmically, a crucial advantage for long products such as those appearing in random‑walk or Lyapunov‑exponent computations.

2. Determinant – The differential of det at M is Tr(Com(M)·dM). Assuming rank(M)≥n−1, the smallest valuation among the (n−1) leading Smith invariants of M, denoted v, determines the image of the integer lattice under the differential: π^{v}O_K. Hence the determinant’s precision is directly tied to the sum of the valuations of the first n−1 invariant factors.

3. Characteristic polynomial – The authors introduce a “precision polygon” for a matrix, which bounds the precision of the coefficients of its characteristic polynomial. This polygon always lies above the classical Hodge polygon, and the gap can be quantified using the same lattice‑based analysis. Numerical tests confirm that the lattice approach yields one or two extra accurate digits compared with coordinate‑wise tracking.

4. LU factorization – By differentiating the equations L·U=A and the forward/backward substitution steps, the authors obtain explicit formulas for the propagation of precision through each stage. The lattice method consistently recovers 2–3 additional p‑adic digits over the naive approach.

5. Subspace arithmetic on Grassmannians – The paper extends the lattice framework to the geometry of Grassmannians. Direct and inverse images, sums, and intersections of subspaces are modeled as smooth maps between Grassmannians; their differentials are computed and used to track precision. Experiments demonstrate that, while coordinate‑wise methods may cause severe loss (or even catastrophic failure) in repeated subspace operations, the lattice method maintains stability and can even increase precision in certain configurations.

Throughout, the authors provide concrete propositions (e.g., Proposition 3.1 for matrix multiplication, Proposition 3.2 for determinants) that give exact formulas for the resulting lattices, as well as bounds on the number of “diffuse digits” (Definition 2.3). The theoretical results are complemented by extensive numerical experiments, whose code is publicly released on GitHub (https://github.com/CETHop/padicprec), ensuring reproducibility.

In summary, the paper establishes that representing p‑adic precision by lattices and propagating it via differentials yields optimal, often dramatically better, stability for a wide range of linear‑algebraic computations. This approach not only reduces the loss of significant digits but also uncovers situations where precision can be gained, opening new possibilities for reliable p‑adic algorithms in number theory and arithmetic geometry.