  The article is concerned with polynomials $g(x)$ whose graphs are "partially
packed" between two horizontal tangent lines. We assume that most of the local
maximum points of $g(x)$ are on the first horizontal line, and most of the
local minimum points on the second horizontal line, except several
"exceptional" maximum or minimum points, that locate above or under two lines,
respectively. In addition, the degree of $g(x)$ is exactly the number of all
extremum points $+1$. Then we call $g(x)$ a multipartite Chebyshev polynomial
associated with the two lines. Under a certain condition, we show that $g(x)$
is expressed as a composition of the Chebyshev polynomial and a polynomial
defined by the $x$-component data of the exceptional extremum points of $g(x)$
and the intersection points of $g(x)$ and the two lines. Especially, we study
in detail bipartite Chebyshev polynomials, which has only one exceptional
point, and treat a connection between such polynomials and elliptic integrals.

Classical Analysis and ODEs