Relatively Prime Polynomials and Nonsingular Hankel Matrices over Finite Fields
arXiv:1011.1760 · doi:10.1016/j.jcta.2010.11.005
Abstract
The probability for two monic polynomials of a positive degree n with coefficients in the finite field F_q to be relatively prime turns out to be identical with the probability for an n x n Hankel matrix over F_q to be nonsingular. Motivated by this, we give an explicit map from pairs of coprime polynomials to nonsingular Hankel matrices that explains this connection. A basic tool used here is the classical notion of Bezoutian of two polynomials. Moreover, we give simpler and direct proofs of the general formulae for the number of m-tuples of relatively prime polynomials over F_q of given degrees and for the number of n x n Hankel matrices over F_q of a given rank
10 pages; to appear in the Journal of Combinatorial Theory, Series A
References in corpus (1)
Cited by in corpus (4)
- Block Companion Singer Cycles, Primitive Recursive Vector Sequences, and Coprime Polynomial Pairs over Finite Fields
- On the rank of Hankel matrices over finite fields
- The Probability of Primeness for Specially Structured Polynomial Matrices over Finite Fields with Applications to Linear Systems and Convolutional Codes
- Finding linearly generated subsequences