Random matrix over a DVR and LU factorization

TL;DR

This paper studies LU factorizations of matrices over a discrete valuation ring, showing average bounds on valuations of factors and applying results to develop an efficient algorithm for computing bases of coherent sheaves.

Contribution

It provides new average valuation bounds for LU factors over a DVR and applies these to improve algorithms in algebraic geometry.

Findings

Average valuations of LU factors are bounded.

Application to efficient basis computation of coherent sheaves.

Enhanced understanding of matrix factorizations over DVRs.

Abstract

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.