Title:
Practical aspects of testing the irreducibility of the non-reciprocal part of a 0,1-polynomialSpeaker: Michael Filaseta
Abstract: I will give some background on 0,1-polynomials and non-reciprocal parts and explain how we can use these ideas to obtain information on the factorization of polynomials. We illustrate a new general result by investigating the factorization of polynomials in the particular sequence 1 + x + x^4 + x^9 + ... + x^(n^2). We also discuss results showing that if the exponents in a 0,1-polynomial increase fast enough, then the non-reciprocal part of the polynomial is irreducible.
Title:
Malle's conjecture for Frobenius groups (25 minutes)Speaker: Harsh Mehta
Abstract: We attain upper bounds for the number of degree d algebraic extensions K/k with Galois group G = F X| H <= S_d as the norm of the discriminant N_(k/Q)(d_(K/k)) is bounded above by X --> infinity. Precisely we look at the following cases: if d=|G| we assume that F is abelian and if d=|F|, we assume that G is a Frobenius group with F abelian. Malle made a conjecture about what the asymptotic of this quantity should be as d and G vary. We show that under a conjecture of Cohen and Lenstra, the upper bounds we achieve, match the prediction of Malle.
Title:
Abstract: A distinct covering system is a set of congruences such that each integers satisfy one or more of the congruences and all moduli of the congruences are distinct positive integers. We consider several extreme distinct covering systems and examine the tools used to establish their extreme properties.
There is no seminar this week, but you can see Number Theory talks at the
INTEGERS Conference 2018
in Augusta, GA.
Title:
Abstract:
A second order polynomial sequence is of "Fibonacci-type" ("Lucas-type") if its Binet formula
has a structure similar to that for Fibonacci (Lucas) numbers. Those are known as "generalized Fibonacci polynomials" GFP.
Some known examples are: Fibobacci Polynomials, Pell polynomials, Fermat polynomials, Chebyshev polynomials,
Morgan-Voyce polynomials, Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials, Chebyshev polynomials, Vieta and Vieta-Lucas polynomials.
It is known that the greatest common divisor of two Fibonacci numbers is again a Fibonacci number. It is called the "strong divisibility property". However, this property does not hold for every second order recursive sequence. We give a characterization of GFPs that satisfy the strong divisibility property. We also give formulas to evaluate the gcd of GFPs that do not satisfy the strong divisibility property.
The "resultant" of two polynomials is the determinant of the Sylvester matrix and
the "discriminant" of a polynomial p is the resultant of p and its derivative. In this talk we discuss closed formulas for
the resultant, the discriminant, and the derivatives of Fibonacci-type polynomials and Lucas-type polynomials. As a corollary we
give explicit formulas for the resultant, the discriminant, and the derivative some known examples of GFPs.
Joint work with R. Higuita, N. McAnally, A. Mukherjee and R. Ramirez.
There is no seminar this week. Enjoy Fall break.
Title:
Abstract:
The speaker will talk about a thing we did - something to do with a polynomial sequence and irreducibility. The "we" is Jacob Juillerat, Jeremiah Southwick and the speaker.
There is no seminar this week, but you are encouraged to attend the
Carolina Math Seminar at USC
.
Title:
Abstract:
This talk is about a comp exam problem given in August. The problem was related to the irreducibility of the polynomial of degree n with constant term 1 and with the coefficient of x^k for each k in [1,n] equal to 1/1111…1, where k ones appear in the denominator. On the comp, students were asked to show the polynomial is irreducible in the cases where n is a prime and where n = 2018. In this talk, we discuss a general method for showing that these polynomials and a number of other polynomials are irreducible provided only that n is sufficiently large.
There is no seminar this week.
There is no seminar this week, but you are encouraged to eat a lot of good food. Happy Thanksgiving!
Title:
Abstract:
In a recent paper, Michael Filaseta and Richard Moy showed that the polynomial acquired by truncating the binomial expansion of (x+1)^n has Galois group S_r where r is the truncation location, except for one instance where the Galois group could be PGL_2(5). In this talk, I will outline how we used Maple to show that there are at most finitely many cases where you get Galois group PGL_2(5). In particular we will delve into the pitfalls of Maple and how we circumvented these.
Title:
Abstract:
A *translation surface* is a Riemann surface, obtained by identifying the sides of a polygon in the Euclidean plane by translations. I will give an overview of a combinatorial problem considered by my collaborator Sunrose Shrestha, the "solution" of which is an additive convolution of divisor sums.
Special cases of these additive convolutions arise in the theory of modular forms, and in some cases are the subject of exact identities. Our main theorem is an asymptotic formula for the additive convolution sum in question. Such a formula was proved in 1927 by Ingham (as we learned later), but our proof is simpler and substantially improves on Ingham's error terms. We also discuss a second approach that may yield secondary main terms for some ranges of the parameters.
What really surprised us is that the result was conjectured in 1916 by Ramanujan. (In the same paper as much more famous conjectures!) We feel that Ramanujan could have easily come up with our proof; the audience will be invited to come to their own conclusions.
This is joint work with Robert Lemke Oliver and Sunrose Shrestha.
Title:
Abstract:
Beginning in May, 1977, the speaker began to devote all of his research efforts to proving the approximately 3300 claims made by Ramanujan without proofs in his notebooks. While completing this task a little over 20 years later, with the help, principally, of his graduate students, he began to work with George Andrews on proving Ramanujan's claims from his "lost notebook.” After another 20 years, with the help of several mathematicians, including my doctoral students, Andrews and I think all the claims in the lost notebook have now been proved.
One entry from the lost notebook connected with the famous Dirichlet Divisor Problem remained painfully difficult to prove. Borrowing from Sherlock Holmes, G.N. Watson's retiring address to the London Mathematical Society in November, 1935 was on the "final problem," arising from Ramanujan's last letter to Hardy. Accordingly, we have called this entry the "final problem," because it was the last entry from the lost notebook to be proved. Early this summer, a proof was finally given by Junxian Li, who just completed her doctorate at the University of Illinois, Alexandru Zaharescu (her advisor), and myself. Since I will tell you how I became interested in Ramanujan and his notebooks, part of my lecture will be historical.
PAlmetto Number Theory Series XXXI (at USC)