Currently, the multi-degree sumcheck runs all groups — CPR (degree d, typically 5) and LogUp (degrees 3 and 2) — with shared challenge points. Each group sends its own degree_g evaluations per round, so the total per-round proof contains Σ_g degree_g = d + 7 field elements, and the verifier performs a separate Horner evaluation and sum-check for each group.
An alternative is to pads all groups to max(degree_g) and batch via a random linear combination (like we do for lookup cols/chunks), producing a single combined polynomial of degree d = max(degree_g) per round. This reduces per-round proof elements from d + 7 to just d, and reduces the verifier's per-round work from multiple group checks to a single Horner evaluation of degree d. Or one could also split it into two groups - high degree and one low degree.
Currently, the multi-degree sumcheck runs all groups — CPR (degree
d, typically 5) and LogUp (degrees 3 and 2) — with shared challenge points. Each group sends its own degree_g evaluations per round, so the total per-round proof contains Σ_g degree_g = d + 7 field elements, and the verifier performs a separate Horner evaluation and sum-check for each group.An alternative is to pads all groups to max(degree_g) and batch via a random linear combination (like we do for lookup cols/chunks), producing a single combined polynomial of degree d = max(degree_g) per round. This reduces per-round proof elements from d + 7 to just d, and reduces the verifier's per-round work from multiple group checks to a single Horner evaluation of degree d. Or one could also split it into two groups - high degree and one low degree.