  Let $T_t^{P_2}f(x)$ denote the solution to the linear Schr\"odinger equation
at time $t$, with initial value function $f$, where $P_2 (\xi) = |\xi|^2$. In
1980, Carleson asked for the minimal regularity of $f$ that is required for the
pointwise a.e. convergence of $T_t^{P_2} f(x)$ to $f(x)$ as $t \rightarrow 0.$
This was recently resolved by work of Bourgain, and Du and Zhang. This paper
considers more general dispersive equations, and constructs counterexamples to
pointwise a.e. convergence for a new class of real polynomial symbols $P$ of
arbitrary degree, motivated by a broad question: what occurs for symbols lying
in a generic class? We construct the counterexamples using number-theoretic
methods, in particular the Weil bound for exponential sums, and the theory of
Dwork-regular forms. This is the first case in which counterexamples are
constructed for indecomposable forms, moving beyond special regimes where $P$
has some diagonal structure.
