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Title: The pentagram map, Poncelet polygons, and commuting difference operators
The pentagram map takes a planar polygon $P$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$ . This map is known to interact nicely with Poncelet polygons, that is, polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of Schwartz states that if $P$ is a Poncelet polygon, then the image of $P$ under the pentagram map is projectively equivalent to $P$ . In the present paper, we show that in the convex case this property characterizes Poncelet polygons: if a convex polygon is projectively equivalent to its pentagram image, then it is Poncelet. The proof is based on the theory of commuting difference operators, as well as on properties of real elliptic curves and theta functions.  more » « less
Award ID(s):
2008021
NSF-PAR ID:
10413208
Author(s) / Creator(s):
Date Published:
Journal Name:
Compositio Mathematica
Volume:
158
Issue:
5
ISSN:
0010-437X
Page Range / eLocation ID:
1084 to 1124
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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