suppose we have a polynomial $p(x,y)$ such that $p(x,y) = 1$ whenever $x+y=1$, and such that all coefficients of $p$ are nonnegative. if $n$ is the number of nonzero coefficients and $d$ is the degree, then $d \leq 2n-3$. for example, if the degree is 3, then you have to have at least 3 terms in $p.$ i will talk about proving this bound and also about proving similar bounds in higher dimensions (polynomials in more than two variables). while the statement of the problem above is elementary, the class of polynomials considered appears as a special case of a hard problem in complex analysis.