Explicit min-max polynomials on the disc

Denote by Pi(2)(n+m-1) := {Sigma(0 <= i+j <= n+m-1) c(i,j)x(i)y(j) : c(i,j) is an element of R} the space of polynomials of two variables with real coefficients of total degree less than or equal to n + m - 1. Let b(0), b(1), ... , b(l) is an element of R be given. For n, m is an element of N, n >= l + 1 we look for the polynomial b(0)x(n) y(m) + b(1)x(n-1) y(m+1) + ... + b(l)x(n-1) y(m+1) + q(x, y), q(x, y) is an element of Pi(2)(n+m-1), which has least maximum norm on the disc and call such a polynomial a min-max polynomial. First we introduce the polynomial 2P(n,m)(x, y) = xG(n-1,m)(x, y) + yG(n,m-1)(x, y) = 2x(n) y(m) + q(x, y) and q(x, y) is an element of Pi(2)(n+m-1), where G(n,m)(x, y) := 1/2(n+m) (U(n)(x)U(m)(y) + U(n-2)(x)U(m-2)(y)), and show that it is a min-max polynomial on the disc. Then we give a sufficient condition on the coefficients b(j), j = 0, ... , l, l fixed, such that for every n, m is an element of N, n >= l + 1, the linear combination Sigma(l)(nu=0) b(nu) P(n-nu,m+nu)(x, y) is a min-max polynomial. In fact the more general case, when the coefficients b(j) and l are allowed to depend on n and in. is considered. So far, up to very special cases, min-max polynomials are known only for x(n) y(m), n, m is an element of N(0). (C) 2011 Elsevier Inc. All rights reserved.