  A function g, with domain the natural numbers, is a quasi-polynomial if there
exists a period m and polynomials p_0,p_1,...,p_{m-1} such that g(t)=p_i(t) for
t=i mod m. Quasi-polynomials classically -- and "reasonably" -- appear in
Ehrhart theory and in other contexts where one examines a family of polyhedra,
parametrized by a variable t, and defined by linear inequalities of the form
a_1x_1+...+a_dx_d <= b(t).
  Recent results of Chen, Li, Sam; Calegari, Walker; and Roune, Woods show a
quasi-polynomial structure in several problems where the a_i are also allowed
to vary with t. We discuss these "unreasonable" results and conjecture a
general class of sets that exhibit various (eventual) quasi-polynomial
behaviors: sets S_t of d-tuples of natural numbers that are defined with
quantifiers ("for all", "there exists"), boolean operations (and, or, not), and
statements of the form a_1(t)x_1+...+a_d(t)x_d <= b(t), where a_i(t) and b(t)
are polynomials in t. These sets are a generalization of sets defined in the
Presburger arithmetic. We prove several relationships between our conjectures,
and we prove several special cases of the conjectures. The title is a play on
Eugene Wigner's "The unreasonable effectiveness of mathematics in the natural
sciences''.
