An Analytic Study of the Irreducibility, Monogeniety, and Squarefreeness of Certain Polynomials [HBNI Th279]

Abstract

This thesis provides a study of problems related to the irreducibility and arithmetic properties of certain families of polynomials. In particular, we emphasize generalized ω-Laguarre polynomials and discriminants of a special class of polynomials. We use some classical analytic methods to approach these problems. The work is divided into two parts. In the first part we establish some results on the irreducibility of generalized ω-Laguerre polynomi- als. The principal tools we applied here are the notion of ω-Newton polygon introduced by Ø. Ore [68] and a generalized version of a lemma of M. Filaseta [19], together with some fundamental results from analytic number theory and the theory of Diophantine equations. The second part of the thesis is based on a quantitative estimate in terms of degree and coe!cients for the number of distinct squarefree parts of discriminants of the monic irreducible polynomials tn +c(atk +b)m → Z[t] of degree n. We study these problems in this part and obtain lower bounds for such quantities, using the square sieve method of D. R. Heath-Brown [28]. Furthermore, assuming the abc- conjecture for number fields, we derive a lower bound for the number of polynomials tn + c(atk + b)m → Z[t] that are monogenic with non-squarefree discriminants or have Galois group Sn . Finally, we conclude the thesis by posing some open problems related to the topics discussed above.