Explicit min–max polynomials on the disc

Denote by Πn+m−12≔{∑0≤i+j≤n+m−1ci,jxiyj:ci,j∈R} the space of polynomials of two variables with real coefficients of total degree less than or equal to n+m−1n+m−1. Let b0,b1,…,bl∈Rb0,b1,…,bl∈R be given. For n,m∈N,n≥l+1n,m∈N,n≥l+1 we look for the polynomial b0xnym+b1xn−1ym+1+⋯+blxn−lym+l+q(x,y),q(x,y)∈Πn+m−12, which has least maximum norm on the disc and call such a polynomial a min–max polynomial. First we introduce the polynomial 2Pn,m(x,y)=xGn−1,m(x,y)+yGn,m−1(x,y)=2xnym+q(x,y)2Pn,m(x,y)=xGn−1,m(x,y)+yGn,m−1(x,y)=2xnym+q(x,y) and q(x,y)∈Πn+m−12, where Gn,m(x,y)≔1/2n+m(Un(x)Um(y)+Un−2(x)Um−2(y))Gn,m(x,y)≔1/2n+m(Un(x)Um(y)+Un−2(x)Um−2(y)), and show that it is a min–max polynomial on the disc. Then we give a sufficient condition on the coefficients bj,j=0,…,l,lbj,j=0,…,l,l fixed, such that for every n,m∈N,n≥l+1n,m∈N,n≥l+1, the linear combination ∑ν=0lbνPn−ν,m+ν(x,y) is a min–max polynomial. In fact the more general case, when the coefficients bjbj and ll are allowed to depend on nn and mm, is considered. So far, up to very special cases, min–max polynomials are known only for xnymxnym,n,m∈N0n,m∈N0.