Recall that the is a non-commutative algebra over a field k with two generators x and y satisfying Lie relation [x,y]=xy−yx = 1. In 1968 Jacques Dixmier stated a conjecture as follows: Every non zero homomorphism from A(k)into itself is an isomorphism. Until now, it is not known that Dixmier's conjecture is true or not. Therefore, in 1990, Professor Nguyen Huu Anh try to find a counter example for that statement in some special cases. To support this efforts, he used some new techniques (Groebner bases method) and exploiting a powerful symbolic computing system, Maple. After getting some preliminary results, Prof. Anh propose the following conjecture: Let P and Q be two polynomials of degrees p,q (p ≥ 1 or q ≥ 1) in the Weyl algebra A(k) = k[x,y], k is algebraically closed field. Assume that: + gcd(p,q) < min (p,q)' and + Lie relation [P,Q] = c, c belongs in k. Then, there exists a polynomial u in A(k) of degree d =gcd(p,q), and two polynomials F,G in one indeterminate such that: P(x,y)=F(u(x,y)) & Q(x,y) = G(u(x,y)). In particular c = 0. In 1997, I investigated the above conjecture in the case: + degrees of polynomials determining homomorphism are consecutively 6 & 9, and furthermore, + k is algebraically closed. I obtained the following results: * On the computing aspect, I assert that 1. there is no counterexample for J. Dixmier's conjecture in the case p=6, q=9. 2. Nguyen Huu Anh's conjecture is true for this case. * On the theoretical aspect, I proved that If Nguyen Huu Anh's conjecture is true, then Dixmier's conjecture is also true
