Auxiliary polynomials and height functions.

다국어 초록 (Multilingual Abstract)

We establish two new results in this dissertation. Recent theorems of Dubickas and Mossinghoff use auxiliary polynomials to give lower bounds on the Weil height of an algebraic number alpha under certain assumptions on alpha. We prove a theorem which introduces an auxiliary polynomial for giving lower bounds on the height of any algebraic number. In particular, we prove the following theorem.
Theorem Let N ∈ Z and alpha ∈ Q . If T ∈ Q [x] is such that deg T ≤ N and T(alpha) ≠ 0 then UN,a,T=U n,a,1=-Nha . Our theorem contains, as corollaries, a slight generalization of the above results as well as some new lower bounds in other special cases.
If alpha1,..., alphar are algebraic numbers such that N=i=1ra i≠i=1ra -1i for some integer N, then a theorem of Beukers and Zagier [2] gives the best possible lower bound on i=1rha i where h denotes the logarithmic Weil height. We will extend this result to allow N to be any totally real algebraic number. That is, we establish the following theorem.
Theorem. Suppose alpha1,..., alpha r are non-zero algebraic numbers and N is a totally real algebraic integer. If alpha1 + ··· + alphar = N and a-11 +···+ a-1r ≠ N then i=1rh ai≥1 2log1+5 2 with equality when r = 1 and alpha1 = 1+52 . This result includes a result of Schinzel [14] which gives a lower bound on the height of a totally real algebraic integer.