engleski

On circumradius equation for cyclic polygons

In this talk we illustrate yet another approach to the Robbins problem, especially well suited for obtaining Heron $r$--polynomials. We have discovered that Robbins problem is somehow related to a Wiener--Hopf factorization. We first associate a Laurent polynomial $L_P$ to a cyclic polygon P, which is invariant under similarity of cyclic polygons ( it is a kind of "conformal invariant"). Then there exists a (Wiener-Hopf ) factorization of $L_P$ into a product of two polynomials, $\gamma_{; ; ; +}; ; ; (1/z)$ and $\gamma_{; ; ; -}; ; ; (z)$ , ( in our case it will be $\gamma_{; ; ; -}; ; ; =\gamma_{; ; ; +}; ; ; =:\gamma$) providing a complex realization of P is given. The factorization (i.e. $\gamma(z)$) is then given in terms of the elementary symmetric functions $e_k$ of the vertex quotients, if we regard vertices of (a realization of) P as complex numbers of equal moduli $(=r)$. For $e_k 's$, viewed as the unknowns, we then obtain a system of $n$ quadratic equations, arising from our Wiener-Hopf factorization, with $n-1$ unknowns (note that $e_n$ is necessarily equal to 1 as a product of all the vertex quotients (we call this a "cocycle property" or simply "cocyclicity")). The consistency condition (obtained by eliminating all $e_k, k=1..n-1$) for our "overdetermined" system will then give a relation between the coefficients of our conformal invariant $L_P$, which in turn will be nothing but the equation relating the inverse square radius of P with the side lengths squared.

cyclic polygon ; circumradius ; Wiener Hopf factorization

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