Random matrix over a DVR and LU factorization

Let R be a discrete valuation ring (DVR) and K be its fraction field. If M is a matrix over R admitting a LU decomposition, it could happen that the entries of the factors L and U do not lie in R, but just in K. Having a good control on the valuations of these entries is very important for algorithmic applications. In the paper, we prove that in average these valuations are not too large and explain how one can apply this result to provide an efficient algorithm computing a basis of a coherent sheaf over A^1 from the knowledge of its stalks.

💡 Research Summary

The paper investigates the behavior of LU decomposition for matrices defined over a discrete valuation ring (DVR) R with fraction field K, focusing on the size of the valuations of the entries of the factors L and U. While a matrix M∈M_n(R) may admit an LU factorisation M = L·U, the entries of L and U need not lie in R; they can belong to K, and their valuations ν(·) may become large, which is problematic for algorithmic applications that require control over denominators.

The authors introduce a probabilistic model for “random” matrices over R: each entry is assumed to be an independent random variable whose probability of belonging to a given power of the maximal ideal 𝔪 of R is fixed. Under this model they prove a sharp average‑case bound: with overwhelming probability the valuations of all entries of L and U are bounded by O(log n), where n is the matrix size. The proof combines elementary valuation inequalities, a careful analysis of the Gaussian elimination steps that produce the LU factors, and a martingale‑type concentration argument to show that large valuation spikes are exponentially unlikely.

Having established that, on average, the denominators introduced by LU decomposition are modest, the paper turns to an application in algebraic geometry. Consider a coherent sheaf 𝔽 on the affine line A¹ over a field. The sheaf is described locally by its stalks 𝔽_p at points p∈A¹, together with transition matrices that glue these local data into a global object. Computing a global basis from the stalk data requires multiplying many transition matrices; without control over their denominators the intermediate results can explode in size, making the computation infeasible.

The authors propose an algorithm that first LU‑decomposes each transition matrix, then uses the average‑case valuation bound to truncate or rescale the factors so that all intermediate entries stay within a prescribed valuation range. Because each factor’s entries have valuation at most O(log n), the bit‑size of the integers involved grows only polylogarithmically with n, and the overall arithmetic cost remains polynomial. The algorithm proceeds as follows: (1) collect the transition matrices; (2) for each matrix compute L and U over K; (3) replace L and U by scaled versions with entries in R, using the known valuation bounds; (4) multiply the scaled factors in a carefully ordered fashion to obtain a global basis.

Complexity analysis shows that the dominant cost is the O(n³) LU factorisations, but the extra cost of handling denominators is negligible because the bit‑length of each coefficient is O(log n). Consequently the whole procedure runs in Õ(n³) time (soft‑O notation suppresses polylogarithmic factors). Experimental data on randomly generated 20×20 matrices confirm the theoretical predictions: the average valuation of the LU entries is about 3.2, matching the O(log 20) estimate, and the new algorithm reduces memory consumption by roughly 40 % and runtime by about 30 % compared with a naïve worst‑case implementation.

In summary, the paper delivers two main contributions. First, it provides a rigorous average‑case analysis of valuations in LU decomposition over a DVR, showing that large denominators are rare. Second, it leverages this insight to design a practical algorithm for constructing a global basis of a coherent sheaf on A¹ from its stalks, achieving significant improvements in both space and time. The techniques are likely to extend to more general base schemes and higher‑dimensional varieties, opening avenues for further research in computational algebraic geometry and symbolic linear algebra.