If $p$ is a polynomial with real coefficients and p(x)>0 on [0,1], then $p(x)=\sum c\_{i,j} x^i(1-x)^j$ with $c\_{i,j}$ positive. I know this is true but but I need a proof/reference. Thanks!



Polynomial positive on an interval
See "Polynomials that are positive on an interval" by myself and Bruce Reznick, Trans. Amer. Math. Soc. 352 (2000), 4677-4692. We give a brief history of this problem along along with a bound for the minimum $m$ so that $p$ can be written $p = \sum\_{i=1}^m c\_i x^i (1-x)^{m-i}$ with $c\_i \geq 0$. The bound depends on the minimum of $p$ on $[0,1]$ and the size of the coefficients. The best reference for this (without the bound) is probably P\'olya-Szego.


