Abstract. Our main theorem shows that the regularity of non-degenerate homogeneous prime ideals is not bounded by any polynomial function of the degree; this holds over any field k. In particular, we provide counterexamples to the longstanding Regularity Conjecture, also known as the Eisenbud-Goto Conjecture (1984). We introduce a method which, starting from a homogeneous ideal I, produces a pr...

We consider self-similar Cantor sets Λ ⊂ R which are either homogeneous and Λ− Λ is an interval, or not homogeneous but having thickness greater than one. We have a natural labeling of the points of Λ which comes from its construction. In case of overlaps among the cylinders of Λ, there are some “bad” pairs (τ, ω) of labels such that τ and ω label the same point of Λ. We express how much the co...

The following theorem is proved. THEOREM. Let X be a homogeneous continuum such that Hl(X) ^ 0. Ij'$/ is the collection of maximal terminal proper subcontinua of X, then (1) The collection ff is a monotone, continuous, terminal decomposition ofX, (2) The nondegenerate elements of%? are mutually homeomorphic, indecomposable, cell-like, terminal, homogeneous continua of the same dimension as X, (...