Mathematics – Algebraic Geometry
Scientific paper
2010-06-30
Mathematics
Algebraic Geometry
Latex, 24 pages
Scientific paper
In this paper we discuss a general framework in which we present a new conjecture, due to Wenhua Zhao, the Image Conjecture. This conjecture implies the Generalized Vanishing Conjecture and hence the Jacobian Conjecture. Crucial ingredient is the notion of a Mathieu space: let $k$ be a field and $R$ a commutative $k$-algebra. A $k$-linear subspace $M$ of $R$ is called a Mathieu subspace of $R$, if the following holds: let $f\in R$ be such that $f^m\in M$, for all $m\geq 1$, then for every $g\in R$ also $gf^m\in M$, for almost all $m$, i.e. only finitely many exceptions. Let $A$ be the polynomial ring in $\zeta=\zeta_1, ...,\zeta_n$ and $z_1, ...,z_n$ over $\mathbb C$. The Image Conjecture (IC) asserts that $\sum_i(\partial_{z_i}-\zeta_i)A$ is a Mathieu subspace of $A$. We prove this conjecture for $n=1$. Also we relate (IC) to the following Integral Conjecture: if $B$ is an open subset of $\mathbb R^n$ and $\sigma$ a positive measure, such that the integral over $B$ of each polynomial in $z$ over $\mathbb C$ is finite, then the set of polynomials, whose integral over $B$ is zero, is a Mathieu subspace of $\mathbb C[z]$. It turns out that Laguerre polynomials play a special role in the study of the Jacobian Conjecture.
No associations
LandOfFree
The Amazing Image Conjecture does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The Amazing Image Conjecture, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Amazing Image Conjecture will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-153658