On an irreducibility theorem of I. Schur
On classifying Laguerre polynomials which have Galois group the alternating group
We show that the discriminant of the generalized Laguerre polynomial is a non-zero square for some integer pair , with , if and only if belongs to one of explicitly given infinite sets of pairs or to an additional finite set of pairs. As a consequence, we obtain new information on when the Galois group of over is the alternating group . For example, we establish that for all but finitely many positive integers , the only for which the Galois group of over is is .
On cycles and orbits of polynomial mappings
On cycles of polynomials with integral rational coefficients
On cyclotomic polynomials with coefficients.
On Hall's conjecture
On Hilbert’s solution of Waring’s problem
In 1909, Hilbert proved that for each fixed k, there is a number g with the following property: Every integer N ≥ 0 has a representation in the form N = x 1k + x 2k + … + x gk, where the x i are nonnegative integers. This resolved a conjecture of Edward Waring from 1770. Hilbert’s proof is somewhat unsatisfying, in that no method is given for finding a value of g corresponding to a given k. In his doctoral thesis, Rieger showed that by a suitable modification of Hilbert’s proof, one can give explicit...
On polynomials taking small values at integral arguments
On polynomials taking small values at integral arguments II
On rational and integral roots of a polynomial
On real polynomials without nonnegative roots.
On sequences of numbers and polynomials defined by linear recurrence relations of order 2.
On some polynomials allegedly related to the abc conjecture
On sums of four cubes of polynomials
On ternary inclusion-exclusion polynomials.
On the coefficients of the cyclotomic polynomial
On the divisibility of polynomials with integer coefficients.
On the generalization of the Fibonacci-coefficient polynomials.
On the height of cyclotomic polynomials
On the irreducibility of 0,1-polynomials of the form f(x)xⁿ + g(x)
If f(x) and g(x) are relatively prime polynomials in ℤ[x] satisfying certain conditions arising from a theorem of Capelli and if n is an integer > N for some sufficiently large N, then the non-reciprocal part of f(x)xⁿ + g(x) is either identically ±1 or is irreducible over the rationals. This result follows from work of Schinzel in 1965. We show here that under the conditions that f(x) and g(x) are relatively prime 0,1-polynomials (so each coefficient is either 0 or 1) and f(0) = g(0) = 1, one...