O sedmnáctém Hilbertově problému
On a class of polynomials with integer coefficients.
On a class of ternary inclusion-exclusion polynomials.
On a decomposition of polynomials in several variables
One considers representation of a polynomial in several variables as the sum of values of univariate polynomials taken at linear combinations of the variables.
On a generalization of the Beiter Conjecture
We prove that for every ε > 0 and every nonnegative integer w there exist primes such that for the height of the cyclotomic polynomial is at least , where and is a constant depending only on w; furthermore . In our construction we can have for all i = 1,...,w and any function h: ℝ₊ → ℝ₊.
On a polynomial conjecture of Pál Turán
On a sequence of nonsolvable quintic polynomials.
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