Date of Submission
Open Access Senior Thesis
Best Senior Thesis in Mathematics
Bachelor of Arts
In this thesis, we give estimates on the successive minima of positive semigroups in lattices and ideals in totally real number fields. In Chapter 1 we give a brief overview of the thesis, while Chapters 2 – 4 provide expository material on some fundamental theorems about lattices, number fields and height functions, hence setting the necessary background for the original results presented in Chapter 5. The results in Chapter 5 can be summarized as follows. For a full-rank lattice L ⊂ Rd, we are concerned with the semigroup L+ ⊆ L, which denotes the set of all vectors with nonnegative coordinates in L. Taking a basis X ⊆ L+ for L + and generating its Z≥0-span,we obtain a conicals ub-semigroup S(X) in L . We call the points in L+ \ S(X) the gaps of S(X). We proceed to describe basic properties of these gaps, but the focus of this thesis is on the restrictive successive minima of L+ and L+ \ S(X), for which we produce bounds in the spirit of Minkowski’s successive minima theorem and its recent generalizations. Further, we apply these results to obtain analogous bounds for sub-semigroups of ideals in totally real number fields, whose image under the Minkowski embedding corresponds to L+ for an appropriate lattice L. These bounds are obtained with respect to the Weil height of elements in the number field.
Wang, Siki, "On properties of positive semigroups in lattices and totally real number fields" (2021). CMC Senior Theses. 2717.