We give infinite triangularization and strict results for algebras of operators on infinite-dimensional vector spaces. introduce a class we call Ore-solvable algebras: these are similar to iterated Ore extensions but need not be free as modules over the intermediate subrings. include many examples particular cases, such group polycyclic groups or finite solvable groups, enveloping Lie algebras,...

0. Introduction. Let H be a complex Hilbert space of finite or infinite dimension, and let E be a collection of bounded linear operators on H. We say E is reducible if there exists a subspace of H, closed by definition and different from the trivial subspaces {0} and H which is invariant under every member of E . We call E triangularizable if the set of invariant subspaces under E contains a ma...

Let f : A → A be a polynomial automorphism of dynamical degree δ ≥ 2 over a number field K. (This is equivalent to say that f is a polynomial automorphism that is not triangularizable.) Then we construct canonical height functions defined on A(K) associated with f . These functions satisfy the Northcott finiteness property, and an Kvalued point on A(K) is f -periodic if and only if its height i...

A multibody car system is a non-nilpotent, non-regular, triangularizable and well-controllable system. One goal of the current paper is to prove this obscure assertion. But its main goal is to explain and enlighten what it means. Motion planning is an already old and classical problem in Robotics. A few years ago a new instance of this problem has appeared in the literature : motion planning fo...

We design two deterministic polynomial time algorithms for variants of a problem introduced by Edmonds in 1967: determine the rank of a matrix M whose entries are homogeneous linear polynomials over the integers. Given a linear subspace B of the n×n matrices over some field F, we consider the following problems: symbolic matrix rank (SMR) is the problem to determine the maximum rank among matri...

We design two deterministic polynomial-time algorithms for variants of a problem introduced by Edmonds in 1967: determine the rank of a matrix M whose entries are homogeneous linear polynomials over the integers. Given a linear subspace B of the n× n matrices over some field F, we consider the following problems: symbolic matrix rank (SMR) is the problem to determine the maximum rank among matr...

OF THE DISSERTATION Algorithms for Polycyclic-by-Finite Groups by Gretchen Ostheimer Dissertation Director: Professor Charles C. Sims Let R be a number eld. We present several algorithms for working with polycyclicbynite subgroups of GL(n;R). Let G be a subgroup of GL(n;R) given by a nite generating set of matrices. We describe an algorithm for deciding whether or not G is polycyclic-bynite. Fo...

In this paper we study triangularization of collections of matrices whose entries come from a finite-dimensional division ring. First, we give a generalization of Guralnick's theorem to the case of finite-dimensional division rings and then we show that in this case the reduced trace function is a suitable alternative for trace function by presenting two triangularization results. The first one...