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The article is concerned with polynomials $g(x)$ whose graphs are "partiallypacked" between two horizontal tangent lines. We assume that most of the localmaximum points of $g(x)$ are on the first horizontal line, and most of thelocal 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 allextremum points $+1$. Then we call $g(x)$ a multipartite Chebyshev polynomialassociated with the two lines. Under a certain condition, we show that $g(x)$is expressed as a composition of the Chebyshev polynomial and a polynomialdefined 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 studyin detail bipartite Chebyshev polynomials, which has only one exceptionalpoint, and treat a connection between such polynomials and elliptic integrals.

Bipartite Chebyshev polynomials and elliptic integrals expressible by elementary functions