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Suppose $x,y,z$ are three variables satisfying $yz=zy, zx=xz,xy=yzx$.

  1. Could anyone give me two (non-commutative) polynomials $f$ and $g$ in the above three variables such that the following equality holds: $$ f(x,y,z)\cdot(3z^2+zyx+x^2)=g(x,y,z)\cdot(3+yx+x^2), $$ i.e., the product of $f$ and $3z^2+zyx+x^2$ is equal to the product of $g$ and $3+yx+x^2$?

  2. Are there any computer software to solve problems of this type? Especially, solve $f(x,y,z)(B_0(y,z)z^{2n}+B_1(y,z)z^nx+x^2)=g(x,y,z)(B_0(y,z)+B_1(y,z)x+x^2)$ for any given $B_i(y,z), n$. Especially, what about the case $B_0=3, B_1=y^2$?

Thanks in advance!

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If you are looking to assign actual values to the variables, then $z=1$ will do the job (no non-commutativity required). If you need the equality to hold on the basis of the commutation relations alone, though, it can't be done (unless $f$ and $g$ are equal and constant). The problem is that each (non-constant) term on the LHS (powers of $3z^2+zyx+x^2$) contains a different pure power of $z$ (none of which can cancel each other out), while the RHS (powers of $3+yx+x^2$) has no pure powers of $z$ at all. – mjqxxxx Sep 30 '13 at 6:06
Sorry for the misleading notation, I have edited it. I mean the product in stead of composition of functions – ougao Sep 30 '13 at 12:26
up vote 0 down vote accepted

$(3z^4+yz^4x+x^2)(3+yzx+x^2)(3z^2+yzx+x^2)=(3z^4+yz^3x+x^2)(3z^2+yz^3x+x^2)(3+yx+x^2) $

I got this by computation, but I suspect there should be some explanation for this equality, any idea on this?

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