Ph.D Student Ph.D Thesis Rosenberg Shai Local and Global Value Sets Department of Mathematics ASSOCIATE PROF. Danny Neftin

Abstract

Given a polynomial f with integer coefficients, let fp denote the polynomial obtained by reducing the coefficients of f modulo a prime integer p. We consider two problems, concerning the extent to which f is determined by properties of fp, where p runs over the set of prime integers.

The first problem is to compare the minimal number of preimages of values of f, with the minimal number of preimages of values of fp. As a specific instance of this problem, consider the following question: for which polynomials f, each value of fp has at least 2 preimages, for almost all primes p?

The second problem is the well studied problem of Kronecker conjugate polynomials, where one asks whether a polynomial f is determined by the value sets of fp. Two polynomials f,g are Kronecker conjugate if fp and gp have the same value set for almost every prime integer. One obvious occurrence of Kronecker conjugate polynomials f,g, is when f(X)=g(aX) for some rational numbers a,b. But there are also known nonobvious examples of Kronecker conjugate polynomials.

We discuss a close analogy between the two problems and show that they can be viewed as two complementary instances of the same problem. Each of the two problems translates to an equivalent group theoretic condition on the corresponding monodromy groups of the polynomials.

Solutions to both problems for indecomposable polynomials, that is, polynomials f that cannot be expressed as a composition g h of two nonlinear polynomials g, h, follow from the classification of monodromy groups of indecomposable polynomials. We consider the problems for decomposable polynomials, where a monodromy classification is out of reach, and solve both of the above problems in all cases where f has no decomposition g h with h satisfying any of the following:

1)      h(X) = L1 XpL2 for some linear polynomials L1, L2 and a prime number p.

2)      h(X) = L1 TpL2 for some linear polynomials L1, L2 and an odd prime number p, where Tp denotes the normalized Chebyshev polynomial.

3)      deg h = 4.