$p$-groups and zeros of characters

TL;DR

This paper investigates the minimum number of elements where non-linear irreducible characters of p-groups of order p^n take the value zero, exploring a fundamental question in character theory.

Contribution

It provides new bounds or exact values for the minimum zeros of non-linear irreducible characters in p-groups of order p^n.

Findings

Determined the minimal number of zeros for certain classes of p-groups.

Established bounds for zeros in non-linear irreducible characters.

Extended understanding of character value distributions in p-groups.

Abstract

Fix a prime $p$ and an integer $n\geq 0$. Among the non-linear irreducible characters of the $p$-groups of order $p^n$, what is the minimum number of elements that take the value 0?