**Symbolic
and Exact Linear Algebra over
Rings and Fields**

**Applications
of Computer Algebra**

**
July 23-27, 2021, Virtual, Online**

*Organizers:*

**Mark
Giesbrecht**^{ 1 },
email: mwg@uwaterloo.ca

**Armin
Jamshidpey**^{ 2 },
email: armin.jamshidpey@uwaterloo.ca

**Éric
Schost**^{ 1 },
email: eschost@uwaterloo.ca

^{1}David R. Cheriton
School of Computer Science, University
of Waterloo

^{2}Institute for Quantum Computing, University
of Waterloo

Symbolic and exact linear algebra over various mathematical domains has developed to become a cornerstone of modern computer algebra systems. Over fields, such as finite fields, we now have well developed methods to work with massive systems of sparse linear equations.

Over the ring of integers, there have been many developments with exact integer matrices, canonical forms such as Smith and Hermite, and integer lattices. Polynomial matrices have seen similar developments with respect to order bases and M-Padé approximation.

Algorithmic techniques are less well-developed for matrices over non-commutative rings, such as rings of differential or difference operators. Similarly algorithms for rings with zero divisors and modules over more general commutative rings present many open problems.

All of these are applicable to a wide range of applications of the theory in science and engineering.

The aim of this special session is to gather experts in the area to discuss the recent achievements and potential new directions, in algorithms, mathematics, and implementations/libraries.

Topics of interest include, over a variety of rings and fields:

• Linear algebra over non-commutative rings

• Lattices and ideals

• Sparse matrices

• Symbolic eigenvalue problems

• Canonical forms

• Linear system solving

• Integer matrices

• Computing null ideals

• Applications: to coding theory, cryptography, mathematics

Talks:

• Applications of the Smith massager of a nonsingular integer matrix, Stavros Birmpilis, University of Waterloo

• Exact linear algebra over the complex numbers, Fredrik Johansson, Inria and Institut. Math. Bordeaux

• Deterministic computation of the characteristic polynomial in the time of matrix multiplication, Vincent Neiger, Univ. Limoges, CNRS

• Null ideals of square matrices over residue class rings of PIDs, Roswitha Rissner, University of Klagenfurt

• Fast and Practical Algorithms for Solving Linear Systems over Number Fields, Jayantha Suranimalee, University of Colombo

• Efficient verification for polynomial matrix computations , Daniel Roche, US Naval Academy

• Frobenius Normal Form: what, why, and how to compute , David Saunders, University of Delaware

• TBA, Gilles Villard, ENS de Lyon