Poly counters = LFSRs (Linear Feedback Shift Register).
They can be replicated programatically easily. With an LFSR the existing value is shifted, 2 existing bits have a formula applied (like XOR) which is used as the incoming bit, and another bit is tapped to use in the audio algorithm.
AFAIK with Atari's use of them with TIA and Pokey the audio algorithm might either use the LFSR output directly or AND it with the flip/flop that toggles at AUDF frequency that if selected by AUDC would normally give a pure tone.
There's various length polys, the values generated are predictable and easily reproducable and repeat each 2 to the power of (n-1) samples.
From memory the Pokey polys update every machine cycle but the TIA ones update based on the AUDF value for each voice. The problem with updating indepdendant of AUDF is that in some situations the AUDF divisor will mean you can read the same sample which gives silent output.
Not sure if the exact LFSR algorithms are documented on a webpage but emulator source code would likely have the info required.