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$c_2$ invariant for powers of primes
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.
The 
${{c}_{2}}$ invariant is an arithmetic graph invariant defined by Schnetz. It is useful for understanding Feynman periods. Brown and Schnetz conjectured that the ${{c}_{2}}$ invariant has a particular symmetry known as completion invariance. This paper will prove completion invariance of the ${{c}_{2}}$ invariant in the case where we are over the field with 2 elements and the completed graph has an odd number of vertices. The methods involve enumerating certain edge bipartitions of graphs; two different constructions are needed.
