an:06445156
Zbl 1316.65038
Caruso, Xavier
Random matrices over a DVR and LU factorization
EN
J. Symb. Comput. 71, 98-123 (2015).
00345038
2015
j
65F05 16W60
\(p\)-adic precision; LU factorization; discrete valuation ring; fraction field; algorithm
Summary: Let \(R\) be a discrete valuation ring (DVR) and \(K\) be its fraction field. If \(M\) is a matrix over \(R\) admitting an 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 on 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 \(\mathbb{A}_K^1\) from the knowledge of its stalks.