When a sumcheck polynomial has the form $g(\mathbf{x}) = \widetilde{\mathsf{eq}}(\mathbf{r}, \mathbf{x}) \cdot p(\mathbf{x})$, the prover can exploit the tensor structure of the eq factor. Since $\widetilde{\mathsf{eq}}(\mathbf{r}, \mathbf{x}) = \prod_i (r_i x_i + (1 - r_i)(1 - x_i))$ factors across coordinates, the prover can maintain a running eq-table and fold in one scalar factor per round rather than treating eq as a general bookkeeping table. This "nearly eliminates" the prover cost of the eq factor. See Section 5 of Speeding Up Sum-Check Proving (Extended Version).
More generally, consider a sumcheck over $g(\mathbf{x}) = \sum_i p_i(\mathbf{x}) q_i(\mathbf{x})$ where only some $q_i$ are eq polynomials. The prover can split the sum into an eq-structured group and an unstructured group, apply eq-elimination to the first group, and handle the second group with standard bookkeeping. This pattern arises naturally for example in the incoming batched tensor evaluation algorithm.
When a sumcheck polynomial has the form$g(\mathbf{x}) = \widetilde{\mathsf{eq}}(\mathbf{r}, \mathbf{x}) \cdot p(\mathbf{x})$ , the prover can exploit the tensor structure of the eq factor. Since $\widetilde{\mathsf{eq}}(\mathbf{r}, \mathbf{x}) = \prod_i (r_i x_i + (1 - r_i)(1 - x_i))$ factors across coordinates, the prover can maintain a running eq-table and fold in one scalar factor per round rather than treating eq as a general bookkeeping table. This "nearly eliminates" the prover cost of the eq factor. See Section 5 of Speeding Up Sum-Check Proving (Extended Version).
More generally, consider a sumcheck over$g(\mathbf{x}) = \sum_i p_i(\mathbf{x}) q_i(\mathbf{x})$ where only some $q_i$ are eq polynomials. The prover can split the sum into an eq-structured group and an unstructured group, apply eq-elimination to the first group, and handle the second group with standard bookkeeping. This pattern arises naturally for example in the incoming batched tensor evaluation algorithm.