Abstract
For a given polynomial F ( t ) = ∑ 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 ) = ∑ 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 ⩽ k ⩽ l ⩽ n . As a by product of this result, we give new refinements of the classical variation diminishing property of Bézier curves.
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