923 research outputs found
Images of Locally Finite Derivations of Polynomial Algebras in Two Variables
In this paper we show that the image of any locally finite -derivation of
the polynomial algebra in two variables over a field of
characteristic zero is a Mathieu subspace. We also show that the
two-dimensional Jacobian conjecture is equivalent to the statement that the
image of every -derivation of such that and
is a Mathieu subspace of .Comment: Minor changes and improvements. Latex, 9 pages. To appear in J. Pure
Appl. Algebr
The tame automorphism group in two variables over basic Artinian rings
In a recent paper it has been established that over an Artinian ring R all
two-dimensional polynomial automorphisms having Jacobian determinant one are
tame if R is a Q-algebra. This is a generalization of the famous Jung-Van der
Kulk Theorem, which deals with the case that R is a field (of any
characteristic). Here we will show that for tameness over an Artinian ring, the
Q-algebra assumption is really needed: we will give, for local Artinian rings
with square-zero principal maximal ideal, a complete description of the tame
automorphism subgroup. This will lead to an example of a non-tame automorphism,
for any characteristic p>0.Comment: 10 page
Two Results on Homogeneous Hessian Nilpotent Polynomials
Let and the Laplace operator. A formal power series is said to be {\it
Hessian Nilpotent}(HN) if its Hessian matrix \Hes P(z)=(\frac {\partial^2
P}{\partial z_i\partial z_j}) is nilpotent. In recent developments in [BE1],
[M] and [Z], the Jacobian conjecture has been reduced to the following
so-called {\it vanishing conjecture}(VC) of HN polynomials: {\it for any
homogeneous HN polynomial of degree , we have for any .} In this paper, we first show that, the VC holds
for any homogeneous HN polynomial provided that the projective
subvarieties and of determined by the principal ideals generated by and
, respectively, intersect only at regular
points of . Consequently, the Jacobian conjecture holds for the
symmetric polynomial maps with HN if has no non-zero
fixed point with . Secondly, we show
that the VC holds for a HN formal power series if and only if, for any
polynomial , when .Comment: Latex, 7 page
Stable Tameness of Two-Dimensional Polynomial Automorphisms Over a Regular Ring
In this paper it is established that all two-dimensional polynomial
automorphisms over a regular ring R are stably tame. In the case R is a
Dedekind Q-algebra, some stronger results are obtained. A key element in the
proof is a theorem which yields the following corollary: Over an Artinian ring
A all two-dimensional polynomial automorphisms having Jacobian determinant one
are stably tame, and are tame if A is a Q-algebra. Another crucial ingredient,
of interest in itself, is that stable tameness is a local property: If an
automorphism is locally tame, then it is stably tame.Comment: 18 page
Venereau-type polynomials as potential counterexamples
We study some properties of the Venereau polynomials b_m=y+x^m(xz+y(yu+z^2)),
a sequence of proposed counterexamples to the Abhyankar-Sathaye embedding
conjecture and the Dolgachev-Weisfeiler conjecture. It is well known that these
are hyperplanes and residual coordinates, and for m at least 3, they are
C[x]-coordinates. For m=1,2, it is only known that they are 1-stable
C[x]-coordinates. We show that b_2 is in fact a C[x]-coordinate. We introduce
the notion of Venereau-type polynomials, and show that these are all
hyperplanes, and residual coordinates. We show that some of these Venereau-type
polynomials are in fact C[x]-coordinates; the rest remain potential
counterexamples to the embedding and other conjectures. For those that we show
to be coordinates, we also show that any automorphism with one of them as a
component is stably tame. The remainder are stably tame, 1-stable
C[x]-coordinates.Comment: 15 pages; to appear in J. Pure and Applied Algebr
A note on k[z]-automorphisms in two variables
We prove that for a polynomial equivalent are: (1) is a
-coordinate of , and (2) and
is a coordinate in for some . This solves a special
case of the Abhyankar-Sathaye conjecture. As a consequence we see that a
coordinate which is also a -coordinate, is a
-coordinate. We discuss a method for constructing automorphisms of
, and observe that the Nagata automorphism occurs naturally as the
first non-trivial automorphism obtained by this method - essentially linking
Nagata with a non-tame -automorphism of , where .Comment: 8 page
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