SPACES OF POLYNOMIALS RELATED TO MULTIPLIER MAPS

Open Access
Author:
Yang, Zhaoning
Graduate Program:
Mathematics
Degree:
Master of Arts
Document Type:
Master Thesis
Date of Defense:
April 06, 2015
Committee Members:
  • Yuriy G Zarkhin, Thesis Advisor
Keywords:
  • polynomial space
  • dimension
  • multiplier maps
Abstract:
Let f be a complex polynomial of degree n. We attach to f a polynomial space W(f) which consists of all complex polynomials p(x) of degree at most n − 2 such that f (x) divides f′′(x)p(x)−f′(x)p′(x). The space W(f) arises for its importance in Yuriy G. Zarkhin’s solution towards a question posed by Yu. S. Ilyashenko. In this paper, we establish an equivalent condition on f (x) that guarantees W (f ) to be nontrivial. Moreover we investigate the dimension of space W (f ) using three independent approaches. The first one uses Hermite interpolation, the second one applies Chinese reminder theorem, the third one invokes combinatorial tools and linear algebra.