Affine k-dimensional intersection conjecture

Prove that, for degree-d polynomial spaces W of dimension r in m affine variables whose common zero set has dimension at most k, the maximum number e^A_r(d,m,k;q) of rational points equals the explicitly defined quantity f^A_r(d,m,k;q) for 0 ≤ k ≤ m−1, m−k ≤ r ≤ \binom{m+d}{d}, and q ≥ d+1.

Background

The affine quantity e^A_r(d,m,k;q) maximizes the number of rational points in the common zero set of an r-dimensional space of affine polynomials of degree at most d, subject to the geometric-dimension constraint dim Z(W) ≤ k. The proposed expression f^A_r(d,m,k;q) is defined through lexicographic ordering of exponent vectors in a bounded monomial box with d-bounds in m−k coordinates and q-bounds in the remaining k coordinates.

The paper constructs examples attaining f^A_r(d,m,k;q), thereby proving it is a lower bound for e^A_r(d,m,k;q). It also notes that the complete-intersection case r=m−k is known, while the general equality remains conjectural.