On the degree of an irreducible factor of the Bernoulli polynomials
On the Gauss-Lucas'lemma in positive characteristic
If is a polynomial with coefficients in the field of complex numbers, of positive degree , then has at least one root a with the following property: if , where is the multiplicity of , then (such a root is said to be a "free" root of ). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree ) with coefficients in a field of positive characteristic (Sudbery's Conjecture). In this paper it is shown that,...
On the Győry-Sárközy-Stewart conjecture in function fields
We consider function field analogues of the conjecture of Győry, Sárközy and Stewart (1996) on the greatest prime divisor of the product for distinct positive integers , and . In particular, we show that, under some natural conditions on rational functions , the number of distinct zeros and poles of the shifted products and grows linearly with if . We also obtain a version of this result for rational functions over a finite field.
On the irreducibility of certain polynomials with coefficients as products of terms in an arithmetic progression
On the irreducible factors of a polynomial over a valued field
We explicitly provide numbers , such that each irreducible factor of a polynomial with integer coefficients has a degree greater than or equal to and can have at most irreducible factors over the field of rational numbers. Moreover, we prove our result in a more general setup for polynomials with coefficients from the valuation ring of an arbitrary valued field.
On the Jacobian conjecture and the configurations of roots.
On the Mulitplicity of Zeros of Polynomials over Arbitrary Finite Dimensional K-Algebras.
On the number of real roots of a solvable polynomial
On the number of terms of a composite polynomial
On the number of terms of a power of a polynomial
On the representation of cyclotomic polynomials as sums of squares
On the zeros of polynomials over arbitrary finite dimensional algebras.
On trinominals of type x...+Ax...+1.