A result on the $c_2$ invariant for powers of primes

TL;DR

This paper establishes a modular relation between the $c_2$ invariant at prime powers, providing evidence for a conjecture in quantum field theory related to arithmetic graph invariants.

Contribution

It introduces a new relation modulo $p$ between the $c_2$ invariants at different prime powers, advancing understanding of their arithmetic properties.

Findings

Relation between $c_2$ invariants at $p$ and $p^s$

Supports Schnetz's conjecture on $c_2$ invariants

Provides algebraic relations for coefficients of polynomial powers

Abstract

The $c_2$ invariant is an arithmetic graph invariant related to quantum field theory. We give a relation modulo $p$ between the $c_2$ invariant at $p$ and the $c_2$ invariant at $p^s$ by proving a relation modulo $p$ between certain coefficients of powers of products of particularly nice polynomials. The relation at the level of the $c_2$ invariant provides evidence for a conjecture of Schnetz.