Separability and factoring polynomials

Factoring Polynomials

Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent of computers and the resultant development of error-correcting codes, factoring over finite fields (e.g., Zp, for p a prime number) has become important as well. To understand this discussion, you need to know what polynomials are, and how to ...

Factoring Modular Polynomials

This paper gives an algorithm to factor a polynomial f (in one variable) over rings like Z=rZ for r 2 Z or F q y]=rF q y] for r 2 F q y]. The Chinese Remainder Theorem reduces our problem to the case where r is a prime power. Then factorization is not unique, but if r does not divide the discriminant of f , our (probabilistic) algorithm produces a description of all (possibly exponentially many...

Factoring Peak Polynomials

Let Sn be the symmetric group of permutations π = π1π2 · · ·πn of {1, 2, . . . , n}. An index i of π is a peak if πi−1 < πi > πi+1, and we let P (π) denote the set of peaks of π. Given any set S of positive integers, we define PS(n) = {π ∈ Sn : P (π) = S}. Burdzy, Sagan, and the first author showed that for all fixed subsets of positive integers S and sufficiently large n we have |PS(n)| = pS(n...

Factoring distance matrix polynomials

In this paper we prove that a vertex-centered automorphism of a tree gives a proper factor of the characteristic polynomial of its distance or adjacency matrix. We also show that the characteristic polynomial of the distance matrix of any graph always has a factor of degree equal to the number of vertex orbits of the graph. These results are applied to full k-ary trees and some other problems. ...

Factoring Polynomials over Local Fields