Polynomization of the Chern–Fu–Tang conjecture

Bessenrodt and Ono’s work on additive and multiplicative properties of the partition function and DeSalvo and Pak’s paper on the log-concavity of the partition function have generated many beautiful theorems and conjectures. In January 2020, the first author gave a lecture at the MPIM in Bonn on a conjecture of Chern–Fu–Tang, and presented an extension (joint work with Neuhauser) involving polynomials. Partial results have been announced. Bringmann, Kane, Rolen, and Tripp provided complete proof of the Chern–Fu–Tang conjecture, following advice from Ono to utilize a recently provided exact formula for the fractional partition functions. They also proved a large proportion of Heim–Neuhauser’s conjecture, which is the polynomization of Chern–Fu–Tang’s conjecture. We prove several cases, not covered by Bringmann et. al. Finally, we lay out a general approach for proving the conjecture.