Absolute Constants of the translates of the Koras-Russell threefold

The main results of this dissertation generalize the established (Crachiola (2005)) non-hyperplanarity of the Koras-Russell threefold x+x 2 y+z 2+t 3 = 0 to the non-hyperplanarity of the translates x + x 2 y + z 2 + t 3 = λ (or the level threefolds) over fields k not necessarily algebraically closed and of arbitrary characteristic. Our theorem includes the case of the generic translate as well, i.e., the hypersurface x + x 2 y + z 2 + t 3 = λ over k(λ) where λ is an indeterminate. As in (Crachiola (2005)), we prove that the ring of Absolute Constant of the relevant affine coordinate ring is a univariate polynomial ring over the ground field. It must be pointed out that there is no prior reason for the ring of Absolute Constants to be essentially the same for each translate; especially surprisingly for the generic translate. To the best of our knowledge, there are no results in the existing literature that determine the ring of Absolute Constants of all translates of a non-hyperplanar threefold.