Weakest sufficient conditions for affine-subfamily sufficiency in the 2^ℓ-adic characterization

Identify the weakest sufficient condition under which, for k = ℓ r and a function f: F_2^n → Z_{2^k} with 2^ℓ-adic decomposition f(x) = ∑_{j=0}^{r-1} 2^{jℓ} c_j(x) where each c_j: F_2^n → Z_{2^ℓ}, verifying only the affine subfamily f_β(x) = c_{r-1}(x) + ∑_{j=0}^{r-2} β_j c_j(x) for β ∈ (Z_{2^ℓ})^{r-1} suffices to conclude the same landscape characterization as obtained by the full family f_F(x) = c_{r-1}(x) + 2^{ℓ-m} F(c_0(x),…,c_{r-2}(x)) with F: (Z_{2^ℓ})^{r-1} → Z_{2^m}, 2 ≤ m ≤ ℓ. Determine the minimal structural assumptions on the lower components c_0,…,c_{r-2} and/or on the partition coefficients that guarantee this implication, given that such a sufficiency result is impossible in full generality.

Background

The paper introduces a 2^ℓ-adic decomposition f(x) = ∑{j=0}^{r-1} 2^{jℓ} c_j(x) for functions f: F_2^n → Z{2^k} with k = ℓ r. It proves an unconditional necessity result and a sufficiency characterization for landscape functions via a small subset of the full family f_F(x) = c_{r-1}(x) + 2^{ℓ-m} F(c_0(x),…,c_{r-2}(x)), where F: (Z_{2^ℓ})^{r-1} → Z_{2^m}.

The authors also develop sufficiency results for the affine subfamily f_β(x) = c_{r-1}(x) + ∑{j=0}^{r-2} β_j c_j(x) under additional structural assumptions (e.g., the common-argument hypothesis). Example 5.6 shows that without such assumptions, gbentness (or landscape) of all fβ does not imply the property for f, demonstrating impossibility in full generality.

The open problem seeks the weakest additional assumptions under which checking only the affine subfamily {f_β} guarantees the same landscape characterization as the full family {f_F}, thus bridging verification efficiency and theoretical completeness.