Hilbert-Kunz functions in a family: Line-\(S_4\) quartics (Q1271126)

With the notation of the previous review (Zbl 0932.13010) let \(G\) be a line-\(S_4.\) Then it is shown that the nondegenerate \(G\)-quartics are parametrized by \(F.\) For an appropriate \(G\) each nondegenerate \(G\)-quartic is a constant multiple of \(h_a = az^4 + (x^2 + yz)(y^2 + xz)\), \(a \in F.\) For each \(a\) the author attaches an integer \(l = l(a)\) that determines completely \(e_n(h_a).\) In particular, it turns out that \(c(h_a) = 3 + 4^{-2l}.\) The surprise is the definition of \(l = l(a).\) The author constructs a \(1\)-parameter family of dynamical systems parametrized by \(F.\) Then \(l(a)\) is defined to be an `escape time' for the system corresponding to \(a.\)