Deterministic Genericity and the Computation of homological Invariants

The main goal of this thesis is to discuss the determination of homological invariants of polynomial ideals. Thereby we consider different coordinate systems and analyze their meaning for the computation of certain invariants. In particular, we provide an algorithm that transforms any ideal into strongly stable position if char k = 0. With a slight modification, this algorithm can also be used to achieve a stable or quasi-stable position. If our field has positive characteristic, the Borel-fixed position is the maximum we can obtain with our method. Further, we present some applications of Pommaret bases, where we focus on how to directly read off invariants from this basis. In the second half of this dissertation we take a closer look at another homological invariant, namely the (absolute) reduction number. It is a known fact that one immediately receives the reduction number from the basis of the generic initial ideal. However, we show that it is not possible to formulate an algorithm – based on analyzing only the leading ideal – that transforms an ideal into a position, which allows us to directly receive this invariant from the leading ideal. So in general we can not read off the reduction number of a Pommaret basis. This result motivates a deeper investigation of which properties a coordinate system must possess so that we can determine the reduction number easily, i.e. by analyzing the leading ideal. This approach leads to the introduction of some generalized versions of the mentioned stable positions, such as the weakly D-stable or weakly D-minimal stable position. The latter represents a coordinate system that allows to determine the reduction number without any further computations. Finally, we introduce the notion of β-maximal position, which provides lots of interesting algebraic properties. In particular, this position is in combination with weakly D-stable sufficient for the weakly D-minimal stable position and so possesses a connection to the reduction number.

Quadrupole moments of radium isotopes from the 7p^2P_3/2 hyperfine structure in Ra II

The hyperfine structure and isotope shift of ^{221- 226}Ra and ^{212, 214}Ra have been measured in the ionic (Ra 11) transition 7s^2 S_{1/2} - 7p ^2 P_{3/2} (\lamda = 381.4 nm). The method of on-line collinear fast-beam laser spectroscopy has been applied using frequency-doubling of cw dye laser radiation in an external ring cavity. The magnetic hyperfine fields are compared with semi-empirical and ab initio calculations. The analysis of the quadrupole splitting by the same method yields the following, improved values of spectroscopic quadrupole moments: Q_s(^221 Ra)= 1.978(7)b, Q_s (^223 Ra)= 1.254(3)b and the reanalyzed values Q_s(^209 Ra) = 0.40(2)b, Q_s(^211 Ra) = 0.48(2)b, Q_s(^227 Ra)= 1.58(3)b, Q_s (^229 Ra) = 3.09(4)b with an additional scaling uncertainty of ±5%. Furthermore, the J-dependence of the isotope shift is analyzed in both Ra II transitions connecting the 7s^2 S_{1/2} ground state with the first excited doublet 7p^ P_{1/2} and 7p^ P_{3/2}.