This paper investigates
rational alternations, principally in R2. Rational alternations in Rn generalize the
polynomial alternations studied in the author’s Moiré Phenomena in Algebraic
Geometry: Polynomial Alternations in Rn. Rational alternations, like polynomial
alternations, have the spirit of diffraction gratings, but may possess singularities,
where grating bands flow together. Both alternations carry with them more
information than ordinary varieties. As in the polynomial case, the systems of
varieties making up two rational alternations generate new systems of varieties under
union (or dually, intersection), corresponding to systems of moiré fringes of various
orders. This paper investigates density functions naturally associated with these
fringes, and studies the behavior of the fringes at points of indeterminacy of the
defining functions.
