# Triangular x-basis decompositions and derandomization of linear algebra algorithms over K[x]

@article{Gupta2012TriangularXD, title={Triangular x-basis decompositions and derandomization of linear algebra algorithms over K[x]}, author={Somi Gupta and Soumojit Sarkar and Arne Storjohann and Jonathan Valeriote}, journal={J. Symb. Comput.}, year={2012}, volume={47}, pages={422-453} }

## 38 Citations

Recent progress in linear algebra and lattice basis reduction

- Computer Science, MathematicsISSAC '11
- 2011

This talk introduces basic tools for understanding how to generalize the Lehmer and Knuth-Schönhage gcd algorithms for basis reduction, and considers bases given by square matrices over K[x] or Z, with the notion of reduced form and LLL reduction.

Deterministic Unimodularity Certification and Applications for Integer Matrices

- Computer Science, Mathematics
- 2013

A deterministic method — “double-plus-one” lifting — is presented to compute the highorder residue R as well as a succinct representation of B to give a heuristic, but certified, algorithm for computing the determinant and Hermite normal form of a square, nonsingular integer matrix.

Deterministic Reduction of Integer Nonsingular Linear System Solving to Matrix Multiplication

- Mathematics, Computer ScienceISSAC
- 2019

This work presents a deterministic reduction to matrix multiplication for the problem of linear system solving and gives an algorithm to produce the 2-adic expansion of 2^eA^-1 b up to a precision high enough such that A+1 b over \Q can be recovered using rational number reconstruction.

Computing Matrix Canonical Forms of Ore Polynomials

- Computer Science, Mathematics
- 2017

This thesis presents algorithms to compute canonical forms of non-singular input matrix of Ore polynomials while controlling intermediate expression swell, and uses the recent advances in polynomial matrix computations to describe an algorithm that computes the transformation matrix U such that UA = P.

Faster Algorithms for Multivariate Interpolation With Multiplicities and Simultaneous Polynomial Approximations

- Computer ScienceIEEE Transactions on Information Theory
- 2015

This paper reduces this multivariate interpolation problem to a problem of simultaneous polynomial approximations, which is solved using fast structured linear algebra and improves the best known complexity bounds for the interpolation step of the list-decoding of Reed-Solomon codes, Parvaresh-Vardy codes, and folded Reed- Solomon codes.

Fast Order Basis and Kernel Basis Computation and Related Problems

- Computer Science, Mathematics
- 2013

The use of the average column degrees instead of the commonly used matrix degrees, or equivalently the maximum column degrees, makes the computational costs more precise and tighter, and the shifted minimal bases computed by the algorithms are more general than the standard minimal bases.

Computing Canonical Bases of Modules of Univariate Relations

- Computer Science, MathematicsISSAC
- 2017

The triangular shape of M is exploited to generalize a divide-and-conquer approach which originates from fast minimal approximant basis algorithms and relies on high-order lifting to perform fast modular products of polynomial matrices of the form P F mod M.

Computing Popov Forms of Polynomial Matrices

- Computer Science, Mathematics
- 2012

A Las Vegas algorithm that computes the Popov decomposition of matrices of full row rank is given and it is shown that the problem of transforming a row reduced matrix to Popov form is at least as hard as polynomial matrix multiplication.

Faster Change of Order Algorithm for Gröbner Bases under Shape and Stability Assumptions

- Computer ScienceISSAC
- 2022

The Hermite normal form of that matrix yields the sought lexicographic Gröbner basis, under assumptions which cover the shape position case, which improves upon both state-of-the-art complexity bounds O~(tD2) and O ~(Dω, since ω<3 and t≤D), and confirms the high practical benefit.

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