| Thu, Mar 23, 2023
|
Number Theory Seminar
3:00 PM
MSCS 514 | | Excursions into the factorization of lacunary polynomials
Michael Filaseta, University of South Carolina
Host: Neil Hoffman
| | Abstract: By a lacunary (or sparse) polynomial, we loosely mean a polynomial of high degree with few terms.
The main interest for the talk is in polynomials in one variable over the rationals.
We discuss a brief history of such investigations to help motivate the interest in the factorization of lacunary polynomials
and then move to some recent work of the speaker involving the factorization of polynomials of the form
\[
f_{0}(x) + f_{1}(x) x^{n} + \cdots + f_{r}(x) x^{rn} \in \mathbb Z[x,y],
\]
with
\[
\gcd{}_{\mathbb Z[x]}\big(f_{0}(x), f_{1}(x), \ldots, f_{r}(x) \big) = 1,
\quad
f_{0}(0) \ne 0,
\quad \text{and} \quad
f_{r}(x) \ne 0.
\]
For this part of the talk, we will focus on specific examples and how to use a general approach for obtaining information
on their factorization. One such example is
\begin{align*}
x^{6n} + (x+1) &x^{5n+1} + 2x^{4n} + (x^4 - x^3 - x^2 - 2x - 2)x^{3n-2}
&\quad+ 2x^{2n} + (x+1) x^{n-2} + 1,
\end{align*}
which we will show is irreducible for all sufficiently large $n$. |
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