18 research outputs found
On Galois groups of a one-parameter orthogonal family of polynomials
No full textFor a fixed integer t>1, we show that if t is not equal to 2, a square ≥4, or three times a square, then the discriminant of the generalized Laguerre polynomial L(s/t)n(x) is a nonzero square for at most finitely many pairs (n,s). Otherwise, the discriminant of L(s/t)n(x) is a nonzero square if and only if (n,s) belongs to one of seven explicitly describable infinite sets or to an additional finite set. This extends the results obtained for t=1 by P. Banerjee, M. Filaseta, C. Finch and J. Leidy. As a consequence, if α is a fixed rational number not equal to 1, 3, 5, or a negative integer, then for all but finitely many n, L(α)n(x) has Galois group Sn, thereby refining a previous result of M. Filaseta – T. Y. Lam and F. Hajir. As an illustration, we give for t=2 infinitely many integer specializations (n,s(n)) such that L(s(n)/2)n(x) has Galois group An. For n≤5, the set of rational numbers α for which the discriminant of L(α)n(x) is a nonzero square is explicitly computed by solving certain generalized Pell-like equations
On Galois groups of generalized Laguerre polynomials whose discriminants are squares
No full textIn this paper, we compute Galois groups over the rationals associated with generalized Laguerre polynomials Ln(∝)(x) whose discriminants are rational squares, where n and α are integers. An explicit description of the integer pairs ((n,∝)for which the discriminant of Ln(∝)(x) is a rational square was recently obtained by the author in a joint work with Filaseta, Finch and Leidy. Among these pairs ((n,∝), we show that for 2≤n≤5, the associated Galois group of Ln(∝)(x) is always
On a generalization of a conjecture of Grosswald
No full textWe generalize a conjecture of Grosswald, now a theorem due to Filaseta and Trifonov, stating that the Bessel polynomials, denoted by yn(x), have the associated Galois group Sn over the rationals for each n. We consider generalized Bessel polynomials yn,β(x) which contain interesting families of polynomials whose discriminants are nonzero rational squares. We show that the Galois group associated with yn,β(x) always contains An if β≥0 and n sufficiently large. For β<0 the Galois group almost always contains An. It is further shown that for β<−2, under the hypothesis of the abc conjecture, the Galois group of yn,β(x) contains An for all sufficiently large n. Using these results, an earlier work of Filaseta, Finch and Leidy and the first author concerning the discriminants of yn,β(x), we are able to explicitly describe the instances where the Galois group associated with yn,β(x) is An for all sufficiently large n depending on
On A Conjecture of Pal Turan and Investigations Into Galois Groups of Generalized Laguerre Polynomials
No full textIn this dissertation we consider two problems. The first problem concerns a conjecture of Pal Turan on distance of a polynomial with integer coefficients from irreducible polynomial. Th problem remains open for polynomials with degree greater than 35. A. Schinzel, in 1970, reformulated Turan\u27s conjecture and subsequently proved the same. In the first part of this dissertation we give a refinement of Schinzel\u27s result.
In the second half we investigate the Galois groups associated with the generalized Laguerre polynomials. We are able to classify Laguerre polynomials with the alternating group as the Galois group. We further compute the Galois groups in certain particular cases
An irreducibility question concerning modifications of Laguerre polynomials
No full textThis paper addresses a question recently posed by Hajir concerning the irreducibility of certain modifications F(x) of generalized Laguerre polynomials L(−n−1−r)n(x) where r≥0 is an integer. For a fixed r≥0, we obtain lower bounds C(r) on n in terms of r such that F(x) is irreducible over the rationals for all n≥C(r). Furthermore, for r≤3, it is shown that F(x) is either irreducible or is a product of a linear polynomial and a polynomial of degree n−1. The set of circumstances in which F(x) has a linear factor for r≤3, is completely described
Corrigendum to “On a generalization of a conjecture of Grosswald” [J. Number Theory 216 (2020) 216–241]
No full textWe extend the result of Lemma 4, [1] to the case that e=0 and ℓ=1 which was missing in [1] but used in the proof of Theorem 1, [1]. © 2020 Elsevier Inc
Representation Theory of Finite Groups
No full textWhen we have a finite group we can identify the elements of the group as matrices
over some field, so we can study the properties of the groups by studying the
matrices only.Representation theory is very useful in number theory and to solve
many group theorytical problems. In this thesis I have tried to show some useful
applications representation theory of finite groups
On Certain Algebraic Aspects of a Family of Orthogonal Polynomials
No full text[Abstract not available
On the nearest irreducible lacunary neighbour to an integer polynomial
No full textThere is an absolute constant D0 > 0 such that if f(x) is an integer polynomial, then there is an integer λ with |λ| ≤ D0 such that xn + f(x) + λ is irreducible over the rationals for infinitely many integers n ≥ 1. Furthermore, if deg f ≤ 25, then there is a λ with λ ∈ {−2, −1, 0, 1, 2, 3} such that xn + f(x) + λ is irreducible over the rationals for infinitely many integers n ≥ 1. These problems arise in connection with an irreducibility theorem of Andrzej Schinzel associated with coverings of integers and an irreducibility conjecture of Pál Turán
