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Generalized Chebyshev Symmetric Multivariable Polynomials Associated to Cyclic and Tangential Polygons

Abstract. Finding equations (or formulas) for the area or circumradius of polygons inscribed in a circle in terms of side lengths is a classical subject (cf Ref.1 ). For the area of a triangle we have the famous Heron’ s formula and for cyclic quadrilaterals we have the Brahmagupta’ s formula. A decade ago D. P.Robbins (Ref.3) found the minimal equations , of degree 7 , satisfied by the squared area of cyclic pentagons and hexagons by a method of undetermined coefficients and he wrote the result in a nice compact form. For the circumradius of cyclic pentagons and hexagons he did not publish the formulas because he was not able to put them into a compact form. The method he used could hardly be used for heptagons due to computational complexity of the approach (leading to a system with 143307 equations). In another approach with two collaborators (Ref.2) a concise heptagon/octagon area formula was obtained recently (not long after D.P.Robbins premature death) in the form of a quotient of two resultants (the quotient still hard to be written explicitly because it would have about one million terms-who wants to be a millioner) . This approach uses covariants of binary quintics. It is not clear if this approach could be effectively used for cyclic polygons with nine or more sides. A nice survey on this and other Robbins conjectures is written by I. Pak. In the MCC2005 talk we have presented an intrinsic proof of the Robbins formulas for the area (and also for the circumradius and area times circumradius) of cyclic hexagons based on an intricate direct elimination of diagonals (the case of pentagon was treated in Ref. 5) and using a new algorithm from Ref. 6. In the early stage we used computations with MAPLE (which sometimes lasted several days!). In this talk we shall explain a simple quadratic system, which seems to be new, for the circumradius and area of arbitrary cyclic polygons based on a Wiener-Hopf factorization of a new Laurent polynomial invariant of cyclic polygons. Explicit formulas, of degree 38 , for the squared circumradius (and less explicit for the squared area) of cyclic heptagons /octagons are obtained. By solving our system in certain algebraic extensions we found a compact form of our circumradius heptagon/octagon formulas with remarkably small coefficients . These formulas could be cosidered as a generalized Chebyshev polynomials (cf. Ref.1, where a trigonometric approach was undertaken by considering a rationalization problem for a sine of a sum of angles , but not finalized even for a cyclic pentagon).For tangential polygons the corresponding polynomials are much easier to find. References 1. A.F. Möbius (1828) Über die Gleichungen, mittelst welcher aus der Seiten eines in einen Kreis zu beschriebenden Vielecks der Halbmesser des Kreises un die Flahe des Vielecks gefunden werden, Crelle's J. 3: 5-34. 2. F. Miller Maley, D.P. Robbins, J. Roskies, On the areas of cyclic and semicyclic polygons, math. MG/0407300v1. 3. D.P. Robbins (1994) Areas of polygons inscribed in a circle, Discrete Comput. Geom. 12: 223-236, 1994. 4. I. Pak, The area of cyclic polygons: Recent progress on Robbins conjectures, mini survey. Adv. Applied Math. Vol.34(2005), 690-696. 5. D. Svrtan, D.Veljan and V. Volenec, Geometry of pentagons: From Gauss to Robbins, math. MG/0403503. 6. D.Svrtan (2005) A new approach to rationalization of surds. (submitted). 7. D.Svrtan (2005) Intrinsic proof of Robbins formula for the area of cyclic hexagons. (submitted). 8. D.Svrtan (2005) Equations for the circumradius and area of cyclic polygons via Wiener-Hopf factorization. Computational aspects and some new formulas. (in preparation). 9. V.V.Varfolomeev, Inscribed polygons and Heron Polynomials, Adv.Sbornik: Mathematics, 194(3):311-331, 2003.

circumradius; area; cyclic polygon; Wiener-Hopf factorization

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