This paper introduces an
object in algebraic geometry akin to but different from an algebraic variety. The main
idea is this: The concept of “inverse image under a polynomial map of a point”
(=algebraic variety) is replaced by “inverse image under a polynomial map of a
periodic subset.” In this paper R is the groundfield, and the periodic subset is
taken to be [0,1) + 2Z ⊆ R. These inverse images, which we call polynomialalternations, are, in R2, like diffraction gratings encountered in optics. They
are closed under complementation as well as “mod two sum.” This sum is
like intersection for ordinary varieties in at least one important way — an
analogue of the usual dimension theorem holds under mod two sum. Union and
intersection are dual, and each gives rise to a phenomenon not encountered with
ordinary varieties — namely striations, or “moiré fringes” are formed. These
fringes run along algebraic varieties, and these varieties correspond to linear
combinations of the polynomials defining the alternations. A density is induced in
each algebraic variety, and this natural density is itself periodic. It depends
on the coefficients of the linear combination; the author determines this
function.
