On asymptotic critical values of a complex polynomial

Journal of Mathematical Analysis and Applications, 2016

We prove that for any circulant matrix C of size n × n with the monic characteristic polynomial p(z), the spectrum of its (n − 1) × (n − 1) submatrix Cn−1 constructed with first n − 1 rows and columns of C consists of all critical points of p(z). Using this fact we provide a simple proof for the Schoenberg conjecture recently proved by R. Pereira and S. Malamud. We also prove full generalization of a higher order Schoenberg-type conjecture proposed by M. de Bruin and A. Sharma and recently proved by W.S. Cheung and T.W. Ng. in its original form, i.e. for polynomials whose mass centre of roots equals zero. In this particular case, our inequality is stronger than it was conjectured by de Bruin and Sharma. Some Schmeisser's-like results on majorization of critical point of polynomials are also obtained. where equality holds if, and only if, all λj lie on a straight line passing through the origin of the complex plane.

Publications of the Research Institute for Mathematical Sciences

Let S ⊂ C n be a non-singular algebraic set and f : C n → C be a polynomial function. It is well-known that the restriction f | S : S → C of f on S is a locally trivial fibration outside a finite set B(f | S) ⊂ C. In this paper, we give an explicit description of a finite set T ∞ (f | S) ⊂ C such that B(f | S) ⊂ K 0 (f | S) ∪ T ∞ (f | S), where K 0 (f | S) denotes the set of critical values of the f | S. Furthermore, T ∞ (f | S) is contained in the set of critical values of certain polynomial functions provided that the f | S is Newton non-degenerate at infinity. Using these facts, we show that if {f t } t∈[0,1] is a family of polynomials such that the Newton polyhedron at infinity of f t is independent of t and the f t | S is Newton non-degenerate at infinity, then the global monodromies of the f t | S are all isomorphic.

Ergodic Theory and Dynamical Systems, 2000

It is shown that a polynomial with a Cremer periodic orbit has a non-accessible critical point in its Julia set provided that the Cremer periodic orbit is approximated by small cycles. Also, this paper contains a new proof of the Douady–Shishikura inequality for the number of non-repelling cycles of a complex polynomial.

Publications de l'Institut Math?matique (Belgrade)

We extend Aziz and Mohammad's result that the zeros, of a polynomial P (z) = n j=0 a j z j , ta j a j−1 > 0, j = 2, 3,. .. , n for certain t (> 0), with moduli greater than t(n − 1)/n are simple, to polynomials with complex coefficients. Then we improve their result that the polynomial P (z), of degree n, with complex coefficients, does not vanish in the disc |z − ae iα | < a/(2n); a > 0, max |z|=a |P (z)| = |P (ae iα)|, for r < a < 2, r being the greatest positive root of the equation x n − 2x n−1 + 1 = 0, and finally obtained an upper bound, for moduli of all zeros of a polynomial, (better, in many cases, than those obtainable from many other known results).

Journal of Differential Equations, 1986