The practical computation of areas associated with binary quartic forms

We derive formulas for practically computing the area of the region |F (x, y)| ≤ 1 defined by a binary quartic form F (X, Y ) ∈ R[X,Y ]. These formulas, which involve a particular hypergeometric function, are useful when estimating the number of lattice points in certain regions of the type |F (x, y)| ≤ h and will likely find application in many contexts. We also show that for forms F of arbitrary degree, the maximal size of the area of the region |F (x, y)| ≤ 1, normalized with respect to the discriminant of F and taken with respect to the number of conjugate pairs of F (x, 1), increases as the number of conjugate pairs decreases; and we give explicit numerical values for these normalized maxima when F is a quartic form.