Vol. 89, No. 2, 1980

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Moiré phenomena in algebraic geometry: polynomial alternations in Rn

Keith Milo Kendig

Vol. 89 (1980), No. 2, 327–349

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 polynomial alternations, 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.

Mathematical Subject Classification 2000
Primary: 14G30, 14G30
Secondary: 78A45
Received: 8 May 1979
Published: 1 August 1980
Keith Milo Kendig