Explicit min-max polynomials on the disc (Q547876)

Let \({\mathcal P}(x,y)\) be a homogeneous polynomial of two variables of degree \(n+m\geq 1\), and let \(\Pi^2_{n+m-1}\) denote the space of polynomials of two variables with real coefficients of total degree \(\leq n+m-1\). Let \(p^*\in \Pi^2_{n+m-1}\). The polynomial \({\mathcal P}-p^*\) is called min-max polynomial on \({\mathcal D}\) if \[ \max_{(x,y)\in{\mathcal D}}|{\mathcal P}(x,y)-p^*(x,y)|=\min_{p\in \Pi^2_{n+m-1}} \max_{(x,y)\in{\mathcal D}}|{\mathcal P}(x,y)-p(x,y)|. \] This paper gives a sufficient condition for a polynomial to be a min-max polynomial on \({\mathcal D}=\{(x,y)\mid x^2+y^2\leq 1\}\), and the authors find an expression for the minimum deviation. They consider the solution of this problem by writing \({\mathcal P}\) in terms of the following bivariate min-max polynomials: \[ P_{n,m}(x,y):=\tfrac12 xG_{n-1,m}(x,y)+\tfrac12 yG_{n,m-1}(x,y)=x^ny^m+q(x,y), \] where \(q\in \Pi^2_{n+m-1}\) and \(G_{n,m}\) are the Gearhart polynomials.