Number Of Irreducible Factors Research Articles

Gaussian limiting distributions for the number of components in combinatorial structures

Consider the number of cycles in a random permutation or a derangement, the number of components in a random mapping or a random 2-regular graph, the number of irreducible factors in a random polynomial over a finite field, the number of components in a random mapping pattern. These random variables all tend to a limiting Gaussian distribution when the sizes of the random structures tend to infinity. Such results, some old and some new, are derived from two general theorems that cover structures decomposed into elementary “components” in either the labelled or the unlabelled case, when the generating function of components has a singularity of a logarithmic type. The proofs are constructed by combining the continuity theorem for characteristic functions with singularity analysis techniques based on Hankel contours.

On the number of irreducible factors of degreek dividing a given polynomial overGF(q)

On the number of irreducible factors of degreek dividing a given polynomial overGF(q)

A Generalization of Swan's Theorem

Let f and g denote polynomials over the two-element field. In this paper we show that the parity of the number of irreducible factors of x'f + g is a periodic function of n, with period dividing eight times the period of the polynomial f 2(x(g/f)' n(g/f)). This can be considered a generalization of Swan's trinomial theorem [3].

О числе неприводимых факторов данного многочлена над конечным полем

О числе неприводимых факторов данного многочлена над конечным полем