Every  Calculus  student  is  familiar  with  the  classical  Rolle’s  theorem  stating that if a real polynomial  p satisfies  p(−1) = p(1),  then it  has a critical  point  in  (−1, 1). In 1934, L. Tschakaloff strengthened this result by finding a minimal interval, contained in (−1, 1), that holds a critical point of every real polynomial    with p(−1) = p(1), up to a fixed degree.  In 1936, he expressed a desire to find an analogue of his result for complex  polynomials.

This talk will present the following Rolle’s theorem for complex polynomials. If p(z) is  a  complex  polynomial  of  degree  n ≥ 5,  satisfying  p(−i)  =  p(i),  then there is at least one critical point of p in the union D[−c; r] ∪D[c; r] of two closed disks with centres −c, c and radius r, where c = cot(2π/n),    r = 1/ sin(2π/n).

If  n = 3,  then  the  closed  disk  D[0; 1/√3]  has  this  property;  and  if  n = 4  then the union of the closed disks D[−1/3; 2/3] ∪D[1/3; 2/3] has this property.  In the last two cases, the domains are minimal, with respect  to  inclusion,  having  this property.

This theorem is stronger than any other known Rolle’s Theorem for complex polynomials of any degree. A minimal Rolle’s domain are found for polynomials of degree 3 and 4, answering Tschakaloff’s  question.

This is a joint work with Blagovest Sendov from the Bulgarian Academy of Sciences.