tailieunhanh - Báo cáo toán học: "the Pentagram Integrals on Inscribed Polygons"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: The Pentagram Integrals on Inscribed Polygons. | The Pentagram Integrals on Inscribed Polygons Richard Evan Schwartz Serge Tabachnikov Submitted Oct 19 2010 Accepted Aug 20 2011 Published Sep 2 2011 Mathematics Subject Classification 37J35 Abstract The pentagram map is a completely integrable system defined on the moduli space of polygons. The integrals for the system are certain weighted homogeneous polynomials which come in pairs El O2 E2 O2 . In this paper we prove that Ek Ok for all k when these integrals are restricted to the space of polygons which are inscribed in a conic section. Our proof is essentially a combinatorial analysis of the integrals. 1 Introduction The pentagram map is a geometric iteration defined on polygons. This map is defined over any field but it is most easily described for polygons in the real projective plane. Geometrically the pentagram map carries the polygon P to the polygon Q as shown in Figure 1. Figure 1 The pentagram map Supported by . Research Grant DMS-0072607 THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P171 1 The first reference we know to some version of the pentagram map is M where it is studied for pentagons. The first author of this paper wrote a series of papers S1 S2 and S3 about the map as defined for general n-gons. See also the recent papers B G oSt1 OST2 Sol and ST . The pentagram map is always defined for convex polygons and generically defined for all p olygons. The pentagram map commutes with projective transformations and induces a generically defined map T on the space Qn of cyclically labelled 1 n-gons modulo projective transformations. T is periodic for n 5 6 but not periodic for n 7. In S3 we introduced a larger space Pn of so-called twisted n-gons and then produced polynomials Oi . O n 2 On Ei . E n 2 En Pn R which are invariant under the pentagram map. We call these polynomials the monodromy invariants. These invariants restrict to give invariants on Qn. See 2 for all relevant definitions. The purpose of this paper is to prove the following .