Nicolae Ciprian Bonciocat

On

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.

D(-1)

The famous irreducibility criteria of Schönemann–Eisenstein and Dumas rely on information on the divisibility of the coefficients of a polynomial by a single prime number. In this paper, we will use some results and ideas of Dumas to provide several irreducibility criteria of Schönemann–Eisenstein–Dumas-type for polynomials with integer coefficients, criteria that are given by some divisibility conditions for their coefficients with respect to arbitrarily many prime numbers. A special attention will be paid to those irreducibility criteria that require information on the divisibility of the coefficients by two distinct prime numbers.

-Quadruples

We provide irreducibility conditions for polynomials of the form f(X) + pkg(X), with f and g relatively prime polynomials with integer coefficients, deg f < deg g, p a prime number and k a positive integer. In particular, we prove that if k is prime to deg g - deg f and pk exceeds a certain bound depending on the coefficients of f and g, then f(X) + pkg(X) is irreducible over ℚ.

Schönemann–Eisenstein–Dumas-Type Irreducibility Conditions that Use Arbitrarily Many Prime Numbers

We provide some square-free criteria for primitive polynomials over unique factorization domains, which do not make use of derivatives or discriminants. Using some ideas of Ostrowski we establish nonvanishing conditions for determinants of matrices with polynomial entries and deduce square-free criteria for polynomials in several variables.

IRREDUCIBILITY CRITERIA FOR SUMS OF TWO RELATIVELY PRIME POLYNOMIALS

We use someclassicalestimatesfor polynomialroots to provide several irreducibilitycriteria for polynomials with integer coefficients that have one sufficiently large coefficient and take a prime value.

Square-free criteria for polynomials using no derivatives

We provide irreducibility conditions for some classes of multivariate polynomials over a field K, namely for polynomials of the form f + pg, where f, g ∈ K[X1, . . . , Xr], degr f < degr g, p ∈ K[X1, . . . , Xr−1] is irreducible over K(X1, . . . , Xr−2), and k ≥ 1 is an integer. More precisely, we prove that if f and g regarded as polynomials in Xr with coefficients in K[X1, . . . , Xr−1] are relatively prime over K(X1, . . . , Xr−1), k is prime to degr g − degr f , and degr−1 p is sufficiently large, then the polynomial f + pg is irreducible over K(X1, . . . , Xr−1).

The Irreducibility of Polynomials That Have One Large Coefficient and Take a Prime Value

We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form f(X)(Y − f 1(X))…(Y − f n (X)) + g(X), with f, f 1, ⋅, f n , g univariate polynomials over an arbitrary field.

Irreducibility criteria for sums of two relatively prime multivariate polynomials

We obtain explicit lower bounds for the Mahler measure for nonreciprocal polynomials with integer coefficients satisfying certain congruences.

Some Pólya-Type Irreducibility Criteria for Multivariate Polynomials

Abstract We provide irreducibility criteria for multiplicative convolutions of polynomials with integer coefficients, that is, for polynomials of the form hdeg f · f(g/h), where f, g, h are polynomials with integer coefficients, and g and h are relatively prime. The irreducibility conditions are expressed in terms of the prime factorization of the leading coefficient of the polynomial hdeg f · f(g/h), the degrees of f, g, h, and the absolute values of their coefficients. In particular, by letting h = 1 we obtain irreducibility conditions for compositions of polynomials with integer coefficients.

CONGRUENCES AND LEHMER'S PROBLEM

We provide explicit upper bounds for the multiplicities of the irreducible factors for some classes of polynomials in two variables X, Y over a field K, regarded as polynomials in Y with coefficients in K[X] whose degrees satisfy certain inequalities. We then obtain similar results for polynomials in an arbitrary number of variables over K.