Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

We prove analogues of several well-known results concerning rational maps between quadrics for the class of so-called quasilinear p-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric methods which have been successfully applied to the study of projective homogeneous varieties over fields cannot be used. We are therefore forced to take an alternative approach, which is partly facilitated by the appearance of several non-traditional features in the study of these objects from an algebraic perspective. Our main results were previously known for the class of quasilinear quadrics. We provide new proofs here, because the original proofs do not immediately generalise for quasilinear hypersurfaces of higher degree.

The position of the polis in the Iliad and the Odyssey is equivocal. Because there is a great span in time between the formal subject material of the poems (ca. 1250 BC) and the time of their composition (ca. 750-700 BC, by convention), the poems present a hybrid world picture with varying and occasionally contradictory systems of social organization. As much as the poems are part of a long epic and formulaic tradition which looks back to an ancestral Mycenaean past, they also have been adapted to the conditions of Homer's lifetime. In political organization, the Iliad and the Odyssey look back through the small tribal groupings and warrior aristocracy of the Dark Ages, to a highly centralized and bureaucratic society governed by divinely ordained kings; but they also look forward to that form of social organization which Homer saw emerging about him — the polis. With this wide disparity in socio-political orientation apparent in the poems, one scholar can say, ‘For Homer and his audience the polis is regarded as the typical form of human community’, whereas another can argue that the significance of the polis is masked and superseded by that of the oikos (the means of social unification preeminent in the Dark Ages). And even if the polis were agreed to be the typical form of human habitation, it is nevertheless argued that the focus in the poems upon the individualistic warrior aristocracy prevents the polis from playing a significant dramatic role in the poems as a higher coercive power: ‘die Stadt habe für die fuhrende Schicht nur an der Peripherie des Daseins gestanden.’ More extensively the same critic writes (p. 164):