Polynomial Mappings and Related Matters

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)." The PI will investigate several aspects of polynomial mappings and related matters. One topic involves orbits of complex polynomials. The proposer will study when the cartesian product of finitely many such orbits has infinitely many points in common with an algebraic subvariety of affine space. This is a dynamical analogue of the Mordell--Lang conjecture about intersections of subgroups and subvarieties of group varieties. Another topic is to determine the polynomials over a field of positive characteristic inducing mappings with certain properties; these properties can be recorded in the monodromy group of the polynomial, so the problem is to determine polynomials with prescribed monodromy group. Via a splitting field construction, polynomials with unusual monodromy groups give rise to curves with large automorphism group. Such polynomials could thus be classified if one could classify curves with large automorphism group over an algebraically closed field of positive characteristic, which is the next topic the PI will study. A fourth topic is the theoretical and algorithmic analysis of the problem of computing the intersection of two subfields of the field of rational functions, along with generalizations to mappings between curves. A final topic addresses connections between discrete mathematics and algebraic geometry, for instance via connecting certain combinatorial objects (Kloosterman sums) with parameter spaces (Shimura curves). This project focuses on polynomials. The classical topic of dynamical systems studies the orbit of a number under a polynomial, which is the collection of numbers obtained by repeatedly applying the polynomial to the number. For instance, the orbit of the number 1 under the polynomial f(x)=3x+1 is 1, 4, 13, 40, and so on. One question is when one such orbit can contain infinitely many numbers whose square is in an orbit of another polynomial. More generally, the PI will study when there is a polynomial relationship between infinitely many points in one orbit and infinitely many in another. The proposer intends to determine all examples of this phenomenon by proving new results about certain types of identities of the form a(b(x))=c(d(x)), where a, b, c and d are rational functions (i.e., ratios of polynomials).