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Communicated by E.M. Friedlander MSC: 33C45 32W99 14R15 a b s t r a c tWe first propose a generalization of the image conjecture Zhao (submitted for publication) [31] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent to a variation of the Mathieu conjecture Mathieu (1997) [21] from integrals of G-finite functions over reductive Lie groups G to integrals of polynomials over open subsets of R n with any positive measures. Via this equivalence, the generalized image conjecture can also be viewed as a natural variation of the Duistermaat and van der Kallen theorem Duistermaat and van der Kallen (1998) [14] on Laurent polynomials with no constant terms. To put all the conjectures above in a common setting, we introduce what we call the Mathieu subspaces of associative algebras. We also discuss some examples of Mathieu subspaces from other sources and derive some general results on this newly introduced notion.

International audienceWe consider the problem of vanishing of the momentsView the MathML sourcemk(P,q)=∫ΩPk(x)q(x)dμ(x)=0,k=1,2,…,Turn MathJax onwith Ω a compact domain in RnRn and P(x)P(x), q(x)q(x) complex polynomials in x∈Ωx∈Ω (MVP). The main stress is on relations of this general vanishing problem to the following conjecture which has been studied recently in Mathieu (1997) [22], Duistermaat and van der Kallen (1998) [17], Zhao (2010) [34] and [35] and in other publications in connection with the vanishing problem for differential operators and with the Jacobian conjecture:Conjecture A.For positive μ if mk(P,1)=0mk(P,1)=0for k=1,2,…k=1,2,… , then mk(P,q)=0mk(P,q)=0for k≫1k≫1for any q.We recall recent results on one-dimensional (MVP) obtained in Muzychuk and Pakovich (2009) [24], Pakovich (2009,2004) [25] and [26], Pakovich (preprint) [28] and prove some initial results in several variables, stressing the role of the positivity assumption on the measure μ. On this base we analyze some special cases of Conjecture A and provide in these cases a complete characterization of the measures μ for which this conjecture holds

The Image Conjecture was formulated by the third author, who showed that it implied his Vanishing Conjecture, which is equivalent to the famous Jacobian Conjecture. We prove various cases of the Image Conjecture and show that how it leads to another fascinating and elusive assertion that we here dub the Factorial Conjecture. Various cases of the Factorial Conjecture are proved.

Motivated by the Mathieu conjecture [Ma], the image conjecture [Z3] and the well-known Jacobian conjecture [K] (see also [BCW] and [E1]), the notion of Mathieu subspaces as a natural generalization of the notion of ideals has been introduced recently in [Z4] for associative algebras. In this paper, we first study algebraic elements in the radicals of Mathieu subspaces of associative algebras over fields and prove some properties and characterizations of Mathieu subspaces with algebraic radicals. We then give some characterizations or classifications for strongly simple algebras (the algebras with no non-trivial Mathieu subspaces) over arbitrary commutative rings, and for quasi-stable algebras (the algebras all of whose subspaces that do not contain the identity element of the algebra are Mathieu subspaces) over arbitrary fields. Furthermore, co-dimension one Mathieu subspaces and the minimal non-trivial Mathieu subspaces of the matrix algebras over fields are also completely determined.A R-subspace M of A is said to be a pre-two-sided Mathieu subspace of A if it is both left and right Mathieu subspace of A. Note