Zero patterns of derivatives of polynomials

Abstract: Motivated by recent work of Nathanson, we study the zero patterns of derivatives of polynomials. For $P$ a polynomial of degree $n$ and $\Lambda=(\lambda_1, \ldots, \lambda_m)$ an $m$-tuple of distinct complex numbers, we consider the $m \times (n+1)$ \emph{dope matrix} $D_P(\Lambda)$ whose $ij$-entry equals $1$ if $P^{(j)}(\lambda_i)=0$ and equals $0$ otherwise (for $1 \leq i \leq m$, $0 \leq j \leq n$). We address several natural questions: When $m$ is $1$ or $2$, what do the possible dope matrices look like, and how many are there? What can we say about general upper bounds on the number of $m \times (n+1)$ dope matrices? For which $m$-tuples $\Lambda$ is the number of $m \times (n+1)$ dope matrices maximized? Does every $\{0,1\}$-matrix appear as the left-most portion of some dope matrix?

Based on joint work with Noga Alon and Kevin O'Bryant.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)