Using set theory to define the number of solutions of polynomials?

[VXXV]

Here, I present a few silly doubts on how to define the maximum number of solutions of a polynomial using set notation and theory.
Let Qn(x) be the inverse of an nth-degree polynomial. Precisely, Qn(x)=|R(x)/Pn(x)|. Here, it is of my interest to define a number, J(n), that provides the maximum number of singular points of  Qn(x) or the maximum number of solutions of Pn(x)=0 since (Qn(x))-1 equals Pn(x). To that end, I have tried to create the following definition: 
J(n)=Sup{\gamma((Qn(x))-1=0):dQ \leq n} in which \gamma((Qn(x))-1=0) denotes the number of solutions of Pn(x)=0 and d indicates the degree of Q. Based on the above, I ask:
1. Is the above definition correct?
2. In set notation, may I use such definition "\gamma((Qn(x))-1=0)" to denote the number of solutions of Pn(x)=0?
3. Is there anything else to consider to define J(n)?

 

 

[studiot]

30 minutes ago, VXXV said:

Here, I present a few silly doubts on how to define the maximum number of solutions of a polynomial using set notation and theory.
Let Qn(x) be the inverse of an nth-degree polynomial. Precisely, Qn(x)=|R(x)/Pn(x)|. Here, it is of my interest to define a number, J(n), that provides the maximum number of singular points of  Qn(x) or the maximum number of solutions of Pn(x)=0 since (Qn(x))-1 equals Pn(x). To that end, I have tried to create the following definition: 
J(n)=Sup{\gamma((Qn(x))-1=0):dQ \leq n} in which \gamma((Qn(x))-1=0) denotes the number of solutions of Pn(x)=0 and d indicates the degree of Q. Based on the above, I ask:
1. Is the above definition correct?
2. In set notation, may I use such definition "\gamma((Qn(x))-1=0)" to denote the number of solutions of Pn(x)=0?
3. Is there anything else to consider to define J(n)?

 

 

I thought that the number of solutions to the general polynomial is always exactly equal to the degree of the polynomial, which in turn is given by MAX(n).

There may, of course be repeated or complex solutions in this reckoning.