A note on retracts of polynomial rings in three variables

In Costa's paper published in 1977, he asks us whether every retract of $k^{[n]}$ is also the polynomial ring or not, where $k$ is a field. In this paper, we give an affirmative answer in the case where $k$ is a field of characteristic zero and $n = 3$.Comment: 4 page

When is each proper overring of R an S(Eidenberg)-domain?

A domain R is called a maximal "non-S" subring of a field L if R [containded in] L, R is not an S-domain and each domain T such that R [containded in] T [contained in or equal] L is an S-domain. We show that maximal "non-S" subrings R of a field L are the integrally closed pseudo-valuation domains satisfying dim(R) = 1, dimv(R) = 2 and L = qf(R)

Dirichlet and Neumann Problems for String Equation, Poncelet Problem and Pell-Abel Equation

We consider conditions for uniqueness of the solution of the Dirichlet or the Neumann problem for 2-dimensional wave equation inside of bi-quadratic algebraic curve. We show that the solution is non-trivial if and only if corresponding Poncelet problem for two conics associated with the curve has periodic trajectory and if and only if corresponding Pell-Abel equation has a solution.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Characterization of Multivariate Permutation Polynomials in Positive Characteristic

Multivariate permutation polynomials over the algebra of formal series over a finite field and its residual algebras are characterized. Some known properties of permutation polynomials over finite fields are also extended.AMS Classification 2000: 13B25, 13F25, 11T55. Keywords: Multivariate permutation polynomials.