Bound the coefficient height of the discriminant-49 Humbert form

Establish an effective bound in terms of n for the coefficient height h(G_n^{n^2}) of the primitive integral exact Humbert form, in particular for h(G_7^49), sufficient to determine the number of primes required for rational reconstruction.

Background

The reconstruction procedure determines the coefficient vector by linear algebra over finite fields and then uses rational reconstruction. The number of primes needed depends on the binary height h(G_n^{n^2}) of the primitive integral modular coefficient vector. The paper derives a Hadamard-type bound depending on the chosen interpolation points, but emphasizes that this bound is very far from sharp: for n=5, the bound is approximately 2.6·10^8 bits, whereas the actual height is 1268 bits.

For n=7, the exact Humbert form has not been computed in the paper, so its height and consequently the required number of primes cannot be predicted in advance. An effective height bound depending only on n would make the computational cost of future reconstructions more predictable.