A characteristic set theory for partial difference polynomial systems is proposed. We introduce the concept of coherent and regular ascending chains and prove that a partial difference ascending chain is the characteristic set of its saturation ideal if and only if it is coherent and regular. This gives a method to decide whether a polynomial belongs to the saturation ideal of an ascending chai...

Suppose that (K, ν) is a valued field, f (z) ∈ K[z] unitary and irreducible polynomial (L, ω) an extension of fields, where L = K[z]/(f (z)). Further suppose A local domain with quotient field K such ν has nonnegative value on positive its maximal ideal, in A[z]. This paper devoted to the problem describing structure associated graded ring gr ω A[z]/(f (z)) for filtration defined by as ν. In pa...

We develop a geometric approach to the study of plane sextics with a triple singular point. As an application, we give an explicit geometric description of all irreducible maximal sextics with a type E7 singular point and compute their fundamental groups. All groups found are finite; one of them is nonabelian.

Received 16 March 2008; accepted 4 May 2008 Abstract: We classify the maximal irreducible periodic subgroups of PGL(q,F) , where F is a field of positive characteristic p transcendental over its prime subfield, q 6= p is prime, and F× has an element of order q . That is, we construct a list of irreducible subgroups G of GL(q,F) containing the centre F×1q of GL(q,F) , such that G/F×1q is a maxim...

Some recent results provide su,cient conditions for complete lattices of closure operators on complete lattices, ordered pointwise, to be pseudocomplemented. This paper gives results of pseudocomplementation in the more general setting of closure operators on mere posets. The following result is 0rst proved: closure operators on a meet-continuous meet-semilattice form a pseudocomplemented compl...

We show that $C^\infty$-surface diffeomorphisms with positive topological entropy have finitely many ergodic measures of maximal in general, and exactly one the topologically transitive case. This answers a question Newhouse, who proved such always exist. To do this we generalize Smale's spectral decomposition theorem to non-uniformly hyperbolic surface diffeomorphisms, introduce homoclinic cla...

Fix a prime p > 2. Let ρ : Gal(Q/Q) → GL2(I) be the Galois representation coming from a non-CM irreducible component I of Hida’s p-ordinary Hecke algebra. Assume the residual representation ρ̄ is absolutely irreducible. Under a minor technical condition we identify a subring I0 of I containing Zp[[T ]] such that the image of ρ is large with respect to I0. That is, Im ρ contains ker(SL2(I0) → SL2...

Let R be a Noetherian commutative ring containing a eld. The test ideal, introduced by Hochster and Huneke in HH1], has emerged as an important object associated to R. The test ideal can be deened as the largest ideal J of R such that JI I for all ideals I of R where I denotes the tight closure of I. Although it is not obvious that a ring R admits a non-zero test ideal, Hochster and Huneke show...