The Berkovich-Dhar conjecture concerns the signs of the coefficients of powers of the polynomial, for a positive
integer 
,
 |
For exponents 
, 5, 6, 7, and 8, it predicts nonnegative coefficients of 
and a single change from positive to negative among the
coefficients of 
, after zero coefficients
are omitted (Berkovich and Dhar 2026).
Jiang (2026b) reported a proof of these sign assertions for all five exponents. The paper also gives the limiting positions of their sign
changes. In the quartic case, if 
is the last index 
with ![[q^(3m+2)]P_n(q)^4>0](/images/equations/Berkovich-DharConjecture/Inline7.svg)
,
then 
as 
(Jiang 2026b). The coefficient notation ![[q^j]P](/images/equations/Berkovich-DharConjecture/Inline10.svg)
denotes the coefficient of 
in 
.
Jiang (2026a) also reported a proof of the modulo-five analog for the square and cube of
 |
For both powers, coefficients with exponents divisible by 5 are strictly positive throughout their range. After zero coefficients are omitted, residue classes 3 and 4 have exactly one change from positive to negative. Residue classes 1 and 2 have the reverse sign pattern by reciprocity.
