Sponsored by the Rutgers University Department of Mathematics and the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

Doron Zeilberger, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu
Nathan Fox, Rutgers University, fox {at} math [dot] rutgers [dot] edu)

Title: Some Recent Results on Polynomials and Polynomiography

Speaker: Bahman Kalantari, Rutgers University

Date: Thursday, September 22, 2016 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ


Polynomials are truly mysterious and beautiful. For many years I have been playing with complex polynomials from an independent point of view, perhaps more as computer scientist than a mathematician. A by-product is Polynomiography: Algorithmic visualization in solving a polynomial equation. The resulting images, called polynomiographs, are not necessarily fractal images. There is a naive misconception that any image coming from iterations must be fractal! Indeed polynomiography enriches fractals, both visually and theoretically. Moreover, it connects polynomials to many other subjects not considered in standard applications. Based on many experiences and interactions with diverse audiences - including perhaps over 100 presentations in more than a dozen countries - there is convincing evidence that polynomiography could become widely popular, leading to novel applications of polynomials in STEM, as well as in art and design. Ironically, formal education is slow in embracing it! This is surprising, especially in view of the general interest in popularizing STEM. I invite mathematicians, STEM educators and students to rethink their notions of polynomials and to explore polynomiography.

In this talk I will highlight some recent results on polynomials and polynomiography from the following:

1. How Many Real Attractive Fixed Points Can a Polynomial Have? We derive an explicit formula for a complex polynomial with a prescribed set of fixed points and corresponding multipliers. Using the formula, we prove a polynomial of degree n can have at most [n/2] fixed points lying on any line in the complex plane. (Arxiv)

2. Solving a Cubic Equation by the Quadratic Formula We prove, a cubic complex polynomial with distinct roots and distinct critical points must have a root whose Voronoi cell contains a critical point. By defining a third order homogeneous recurrence relation at such a critical point, we generate a sequence guaranteed to converge to that root. This gives a new method for solving a cubic equation different from Cardano's formula, easy to remember and maybe more practical. (Arxiv)

3. A Necessary and Sufficient Condition for Local Maxima of Polynomial Modulus Over Unit Disc We give necessary and sufficient condition for a local maximum of polynomial modulus over the unit disc, proving that it is a fixed point of a certain function. In particular, the infinity norm of a polynomial is attained at a point satisfying a convenient formula. The formula suggests iterative methods for computing the infinity norm. We give two such algorithms, including a Newton-like method and present some corresponding polynomiography. (Arxiv)

4. The 3x+1 Polynomials, Their Zeros and Polynomiography To each natural number N satisfying the famous 3x+1 property (conjectured to be valid for any natural number), we associate a unique monic polynomial, called its 3x+1 polynomial, having N as the constant term. The 3x+1 polynomial implies a unique factorization of N in terms of the product of its roots. We using an existing family of bounds on the modulus of zeros of a general polynomial to bound the modulus of the zeros of the 3x+1 polynomial. We give some associated polynomiography and extend the results to Gaussian integers. (Forthcoming)