We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a unique factorization domain of characteristic p > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case m = n this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic.
We present a class of counterexamples to the Cancellation Problem over arbitrary commutative rings, using non-free stably free modules and locally nilpotent derivations.
We study the multiplicative lattices which satisfy the condition for all . Call them sharp lattices. We prove that every totally ordered sharp lattice is isomorphic to the ideal lattice of a valuation domain with value group or . A sharp lattice localized at its maximal elements are totally ordered sharp lattices. The converse is true if has finite character.
Butler groups formed by factoring a completely decomposable group by a rank one group have been studied extensively. We call such groups, bracket groups. We study bracket modules over integral domains. In particular, we are interested in when any bracket -module is tensor a bracket group.
We describe a cluster algebra algorithm for calculating -characters of Kirillov–Reshetikhin modules for any untwisted quantum affine algebra . This yields a geometric -character formula for tensor products of Kirillov–Reshetikhin modules. When is of type , this formula extends Nakajima’s formula for -characters of standard modules in terms of homology of graded quiver varieties.
We define various ring sequential convergences on and . We describe their properties and properties of their convergence completions. In particular, we define a convergence on by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields . Further, we show that is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.
Let F=X-H: → be a polynomial map with H homogeneous of degree 3 and nilpotent Jacobian matrix J(H). Let G=(G1,...,Gn) be the formal inverse of F. Bass, Connell and Wright proved in [1] that the homogeneous component of of degree 2d+1 can be expressed as , where T varies over rooted trees with d vertices, α(T)=CardAut(T) and is a polynomial defined by (1) below. The Jacobian Conjecture states that, in our situation, is an automorphism or, equivalently, is zero for sufficiently large d....
Let F = X + H be a cubic homogeneous polynomial automorphism from to . Let be the nilpotence index of the Jacobian matrix JH. It was conjectured by Drużkowski and Rusek in [4] that . We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.