Let $p(t)=a_0+a_1 t+a_2 t^2+... +a_nt^n$ and $I=[a,b]$.
I have seen an article in which a necessary and sufficient condition was given on the coefficients $a_i$ in the cubic case ($n=3$) for the positivity (nonegativity) on the interval $I=[0,1]$. (Jochen W. Schmidt, Walter Heß: Positivity of cubic polynomials on intervals and positive spline interpolation)
Is there any known result (sufficient condition or necessary and sufficient condition on $a_i$) for more general cases ($n>3$)?
Edit:
I have found an article for $n=4$ too. (Gary Ulrich, Layne T. Watson: Positivity Conditions for Quartic Polynomials)