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Converting arithmetic to things based on polynomials - when I was reading about the Atari 2600's TIA I recall some things about how polynomial counters were implemented internally to avoid having to deal with decimal->binary conversion in the chip (this is an 8-bit ASIC clocked at NTSC rate that generates a TV signal and must be manipulated in near real time by the CPU to generate graphics). Wondering how this could be related/applicable.

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I don't see how decimal-to-binary conversion would have anything to do with the TIA chip. I suspect that it used a polynomial counter instead of a binary integer counter because it used fewer gates and had less delay.

By the way, the CPU, like many others back then, could do addition and subtraction with binary-coded decimals.

To position objects horizontally ideally you would have a register where you load a value between 0 and 160, but instead you have to strobe a register when the CRT beam is near the horizontal position you want, which will be +/- some pixels due to the CPU being 1/3 the clock of the TIA and 6502 instructions consuming 2 to 8 cycles. So then you have to "fine tune" using an additional register to shift it left/right -7 to 8 pixels.

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