Dear Qasim:

"Polynomials" is not my specialty , but I see that the tabs on the topic can be of some help.

Best regards,

Debra

https://www.researchgate.net/topic/Polynomials?ev=tp_pst_dtl_xkey

Thank you dear Debra Sharon Ferdinand-James

Dear Qasim Mohammed Hussein

Thank you for inviting me to participate in the answer to this mathematical question.

Because it is not my field of competence, I will simply follow the opinions and contributions of the experts, hoping to learn from the content of the same.

Jose Luis

Dear Qasim Mohammed Hussein

I will give  one example but I don't know the exact expectation 

the polynomial is of the form

x2 + y2 +z2+4xy+4yz+2zx

the matrix form of this equation is 

1     2      1

2    1      2

1     2     1

now you can able to find the inverse of the matrix by using any one of the matrix methods

I am not expert in this field, my subject is mathematical economics. In an ordinary mathematical sense the existence of inverse function means that the function is one-to-one. Therefore it has to be one variable polynomial and it has to be monotonic. Now just differentiate your function and look at the sign of obtained function - if it does not change then it is the case you need.     

Interesting question....