Mathieu Subspaces And The Image Conjecture

The Mathieu subspace is the natural generalization of ideal, which originates from the study of the Jacobian conjecture. The definition was given by Wenhua Zhao in 2007 and the theory of Mathieu subspaces is at an initial stage at present.In the first part of the thesis, we introduce the definition.background and development of Mathieu subspaces. First of all we describe the relation among Mathieu subspace, image conjecture, (generalized)vanishing conjecture and Jacobian conjecture, then introduce the results about the image conjecture and the generalized vanishing conjecture which have been got. In the second part, we give some results obtained by ourselves about Mathieu subspaces and the generalized vanishing conjecture. Let k is a field, chark= 0, k[x, y] is the binary polynomial algebra on k, we show that the generalized vanishing conjecture holds for the differential operator ∧＝((?)x-Φ((?)y))(?) on k[x,y], where Φ((?)y) is the polynomial about (?)y), that is if Φ(ξ)= qo+q1+q2ξ2+…+qξ, then Φ((?)y)= q0+q1(?)y+q2(?)y2+…+qs(?)ys. In addition, we prove the images of some polynomial derivations are Mathieu subspaces.The main results are given as follow:Theorem 0.1 Vanishing conjecture holds for the differential operator A=((?)x-Φ((?)y))(?)y on Jfc[x,y] and all polynomials P(x,y) ∈ k[x, y], that is if ∧mPm= 0 holds for all m≥ 1, then for arbitrary polynomial Q ∈ k[x,y], ∧m(QPm)= 0 holds for all m＞＞0.Theorem 0.2 Let D:= M ∑in=1 aizi(?)zi be a derivation of n-variables polynomial algebra k[z]:= k[z1,,z2,…,zn], where ai ∈ k(1≤ i≤ n), Mis a monomial of k[z], then the image ImD of D is a Mathieu subspace of k[z].