A Refinement of the Variation Diminishing Property of Bézier Curves

International audienceFor a given polynomial $F(t)=\sum_{i=0}^n p_i B_i^n(t)$, expressed in the Bernstein basis over an interval $[a,b]$, we prove that the number of real roots of $F(t)$ in $[a,b]$, counting multiplicities, does not exceed the sum of the number of real roots in $[a,b]$ of the polynomial $G(t)=\sum_{i=k}^l p_i B_{i-k}^{l-k}(t)$ (counting multiplicities) with the number of sign changes in the two sequences $(p_0,...,p_k)$ and $(p_l,...,p_n)$ for any value $k,l$ with $0\leq k\leq l\leq n$. As a by product of this result, we give new refinements of the classical variation diminishing property of Bézier curves