3 research outputs found
Uniform Probability and Natural Density of Mutually Left Coprime Polynomial Matrices over Finite Fields
We compute the uniform probability that finitely many polynomials over a
finite field are pairwise coprime and compare the result with the formula one
gets using the natural density as probability measure. It will turn out that
the formulas for the two considered probability measures asymptotically
coincide but differ in the exact values. Moreover, we calculate the natural
density of mutually left coprime polynomial matrices and compare the result
with the formula one gets using the uniform probability distribution. The
achieved estimations are not as precise as in the scalar case but again we can
show asymptotic coincidence
Probability estimates for reachability of linear systems defined over finite fields
This paper deals with the probability that random linear systems defined over a finite field are reachable. Explicit formulas are derived for the probabilities that a linear input-state system is reachable, that the reachability matrix has a prescribed rank, as well as for the number of cyclic vectors of a cyclic matrix. We also estimate the probability that the parallel connection of finitely many single-input systems is reachable. These results may be viewed as a first step to calculate the probability that a network of linear systems is reachable