Closed polynomials in polynomial rings over integral domains

Closed polynomials and their applications for computations of kernels of monomial derivations

“…In this section, we study closed polynomials and related topics. Some results in this section are generalizations for some results of papers written by the second and third authors [7], [12] and [13]. Let R be an integral domain and let R[X] = R[x 1 , .…”

“…In section 1, we recall some kinds of derivations and some concepts; coordinates, closed polynomials and factorially closed polynomials over an integral domain. In section 2, we give some results on closed polynomials and factorially closed polynomials which are generalizations of results in [7], [12] and [13]. Moreover, in Example 2.7, we show that Danielewski surface and Koras-Russell threefold are factorially closed polynomials, but they are not coordinates.…”

“…, x n , namely, J(F ) = (∂f i /∂x j ) 1≤i, j≤n . The following proposition is a generalization of [7,Proposition 3.6] to the case where the coefficient ring is an integral domain of characteristic zero and n ≥ 1.…”

“…

In this paper, we give some results on closed polynomials and factorially closed polynomial in n variables which are generalizations of results in [7], [12] and [13]. In particular, we give a characterization of factorially closed polynomials in n variables over an algebraically closed field for any characteristic.

…”

“…On the other hand, the second and third authors studied closed polynomials in [7], [12] and [13]. Here, a non-constant polynomial f ∈ k[X] \ k is a closed polynomial if the ring k[f ] is integrally closed in k[X].…”