Ph.D Thesis | |

Ph.D Student | Rosenberg Shai |
---|---|

Subject | Local and Global Value Sets |

Department | Department of Mathematics |

Supervisor | ASSOCIATE PROF. Danny Neftin |

Given a polynomial f with integer
coefficients, let f_{p} 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 f_{p},
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 f_{p}. As a specific instance of this problem, consider
the following question: for which polynomials f, each value of f_{p}
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 f_{p}. Two polynomials
f,g are Kronecker conjugate if f_{p} and g_{p} 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) = L_{1}
○ X^{p}○L_{2} for some linear
polynomials L_{1}, L_{2} and a prime number p.

2) h(X) = L_{1}
○ T_{p}○L_{2} for some linear
polynomials L_{1}, L_{2} and an odd prime number p, where T_{p}
denotes the normalized Chebyshev polynomial.

3) deg h = 4.