Blum, Cucker, Shub and Smale have shown that the problem \P = NP ?" has the same answer in all algebraically closed elds of characteristic 0. We generalize this result to the polynomial hierarchy: if it collapses over an algebraically closed eld of characteristic 0, then it must collapse at the same level over all algebraically closed elds of characteristic 0. The main ingredient of their proof...

In his paper mentioned in the title, which appears in the same issue of this journal, Mehdi Radjabalipour derives the cyclic decomposition of an algebraic linear transformation. A more general structure theory for linear transformations appears in Irving Kaplansky's lovely 1954 book on infinite abelian groups. We present a translation of Kaplansky's results for abelian groups into the terminolo...

It is well-known that the theory of algebraically closed fields is not finitely axiomatizable. In this note, we prove that the theory of integrally closed integral domains is also not finitely axiomatizable. All rings in this paper are assumed commutative with identity. Let L be the language of rings, that is, the language whose signature consists of equality, binary function symbols + and ·, a...

The speed of a class graphs counts the number on vertex set {1,…,n} inside as function n. In this paper, we investigate for many classes that naturally arise in discrete geometry, example intersection segments or disks plane. While upper bounds follow from Warren's theorem (a variant Milnor and Thom), all previously known lower were obtained ad hoc constructions very specific classes. We prove ...