12 research outputs found
Irreducibility criteria for pairs of polynomials whose resultant is a prime number
We use some classical estimates for polynomial roots to provide several
irreducibility criteria for pairs of polynomials with integer coefficients
whose resultant is a prime number, and for some of their linear combinations.
Similar results are then obtained for multivariate polynomials over an
arbitrary field, in a non-Archimedean setting.Comment: 20 page
ON D(-1)- Quadruples
Quadruples (a; b; c; d) of positive integers a < b < c < d with the property that the product of any two of them is one more than a perfect square are studied. Improved lower and upper bounds for the entries b and c are established. As an application of these results, a bound for the number of such quadruples is obtained
Some factorization results for bivariate polynomials
We provide upper bounds on the total number of irreducible factors, and in
particular irreducibility criteria for some classes of bivariate polynomials
over an arbitrary field . Our results rely on information
on the degrees of the coefficients of , and on information on the
factorization of the constant term and of the leading coefficient of ,
viewed as a polynomial in with coefficients in . In
particular, we provide a generalization of the bivariate version of Perron's
irreducibility criterion, and similar results for polynomials in an arbitrary
number of indeterminates. The proofs use non-Archimedean absolute values, that
are suitable for finding information on the location of the roots of in an
algebraic closure of .Comment: 14 page
Some elementary zero-free regions for Dirichlet series and power series
Adapting some elementary methods used by a number of authors to investigate the location of roots of polynomials with complex coefficients, we present some results which provide zero-free regions for Dirichlet series and power series
ON D(-1)- Quadruples
Quadruples (a; b; c; d) of positive integers a < b < c < d with the property that the product of any two of them is one more than a perfect square are studied. Improved lower and upper bounds for the entries b and c are established. As an application of these results, a bound for the number of such quadruples is obtained
Irreducibility criteria for compositions of polynomials with integer coefficients
International audienc