Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question
1  
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
add comment

1 Answer

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?

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.