On the Galois group of the generalized Fibonacci polynomial.
On the Galois spectra of polynomials with integral parameters.
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 greatest common divisor of two univariate polynomials, II
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 ...-invariant for quadratics forms and the linkage of cyclic algebras.
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...
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 elliptic curves with CM cover large algebraic fields
Elliptic curves with CM unveil a new phenomenon in the theory of large algebraic fields. Rather than drawing a line between and or and they give an example where the line goes beween and and another one where the line goes between and .
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 number of zero trace elements in polynomial bases for F2n.
Let Fq denote the finite field of q elements. O. Ahmadi and A. Menezes have recently considered the question about the possible number of elements with zero trace in polynomial bases of F2n over F2. Here we show that the Weil bound implies that there is such a basis with n + O(log n) zero-trace elements.
On the reducibility type of trinomials
On the representation of cyclotomic polynomials as sums of squares
On the roots of polynomials over division algebras.
On the set of distances between two sets over finite fields.