Computing the elementary symmetric polynomials of the multiplier spectra of the maps 𝑧²+𝑐

Authors

Grayson Jorgenson, Florida Institute of Technology

Abstract

Let f be a complex quadratic rational map. The ith elementary symmetric polynomial of the formal n multiplier spectra of f is denoted &sigmai(n)(f). The values of these polynomials are invariant under conjugation by the projective linear group and are interesting to the study of the moduli space of quadratic rational maps. For every positive integer n and i in the appropriate range, &sigmai(n)(f) is in Z[&sigma1, &sigma2] where &sigma1, &sigma2 are &sigma1(1)(f, &sigma2(1)(f), respectively. Despite this, the &sigmai(n)(f) are difficult to compute. By restricting our focus to the family of quadratic polynomials z2+c, computations become simpler. We determine an upper bound for the degrees of the &sigmai(n) for the maps of the form z2+c by arguing in terms of the growth rates of their periodic points and corresponding multipliers. We also include computations of the forms of the &sigmai(n) for n =2,…,6 for these maps.

Author Bio

Grayson Jorgenson graduated in 2015 from the Florida Institute of Technology with a B.S. in applied mathematics. He completed this work during his senior year there. Currently he is a pure mathematics graduate student at Florida State University and is especially interested in algebra, topology, and algebraic geometry. Outside of math, he enjoys playing piano and programming.

Faculty Sponsor

Benjamin Hutz

Recommended Citation

Jorgenson, Grayson (2015) "Computing the elementary symmetric polynomials of the multiplier spectra of the maps 𝑧²+𝑐," Rose-Hulman Undergraduate Mathematics Journal: Vol. 16: Iss. 2, Article 12.
Available at: https://scholar.rose-hulman.edu/rhumj/vol16/iss2/12