Height needed for boundedly many guards

Determine the least asymptotic growth of a coefficient-height function H(B) such that the degree-d arithmetic guard number of the unit square for rational targets of projective height at most B remains bounded, and determine how the answer changes for other polygons and target fields.

Background

For degree-one carriers, the paper proves that keeping the number of guards bounded for rational target-height slices requires coefficient height of order B, and that this scale is sufficient. It also establishes that degree-one edge guarding of quadratic targets can require height of order B².

The unresolved higher-degree problem asks for the optimal height-growth threshold when d≥2. The question concerns boundedly many point guards in the unit square and whether polygonal geometry or the arithmetic of the target field changes that threshold.