I May I use set theory to define the number of solutions of polynomials?

V9999
Messages
17
Reaction score
3
TL;DR Summary
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 ##Q_{n}(x)## be the inverse of an nth-degree polynomial. Precisely,

$$Q_{n}(x)=\displaystyle\frac{1}{P_{n}(x)}$$,

It is of my interest to use the set notation to formally define a number, ##J_{n}## that provides the maximum number of solutions of ##Q_{n}(x)^{-1}=0##. Despite not knowing how to proceed, below you may find my attempt.

Let the maximum number of solutions of ##Q^{-1}_{n}(x)=0## be

$$J_{n}=\text{Sup}\{\pi(Q^{-1}_{n}(x)=0):\partial Q_{n}^{-1}(x) \leq n\}$$,

in which ##\partial## denotes "the degree of" and ##\pi(Q^{-1}_{n}(x)=0)## is the number of solutions of ##Q^{-1}_{n}(x)=0##.

Based on the above, I ask:

1. Is the above definition correct?
2. How may improve and formally define ##J_{n}## using proper notation of set theory and mathematics? That is to say, is there anything else that I should consider to define ##J_{n}## in the way stated above?
3. In order to define the degree of a polynomial, should I consider ##n \in \mathbb{N}## or ##n \in \mathbb{Z}##?

Thanks in advance.
 
Physics news on Phys.org
Why are we even talking about Q instead of P?
 
  • Like
Likes V9999, PeroK, martinbn and 1 other person
Why not just write that ##J_n=n##?
 
##\frac {1}{P_n(x)} ## here, will never be ##0## on the Reals, if I understood you correctly, except possibly in the limit, when ## P_n## grows without bound. Notice you first wrote ##Q_n(x)^{-1}## and then you used ##Q_n^{-1}(x)##.
For your second question, the degree of a polynomial may be 0 or a Natural number; never a negative Integer.
 
WWGD said:
##\frac {1}{P_n(x)} ## here, will never be ##0## on the Reals, if I understood you correctly, except possibly in the limit, when ## P_n## grows without bound. Notice you first wrote ##Q_n(x)^{-1}## and then you used ##Q_n^{-1}(x)##.
For your second question, the degree of a polynomial may be 0 or a Natural number; never a negative Integer.
In his notations ##Q^{-1}## is just ##P##, uless he means inverse function, which i doubt because it wouldnt exist.
 
  • Like
Likes V9999 and WWGD
Office_Shredder said:
Why are we even talking about Q instead of P?
Hi, Office_Shredder. I hope you are doing well.
First, thanks for commenting. Second, I could be P since ##(Q_{n}(x))^{-1}## is ##P_{n}##. In light of the foregoing, the definition would be ##Sup\{\pi(P_{n}(x)=0):\partial P \leq n\}##. Based on the above, is there anything else that I should consider to define ##J_{n}##? Thanks in advance.
 
WWGD said:
##\frac {1}{P_n(x)} ## here, will never be ##0## on the Reals, if I understood you correctly, except possibly in the limit, when ## P_n## grows without bound. Notice you first wrote ##Q_n(x)^{-1}## and then you used ##Q_n^{-1}(x)##.
For your second question, the degree of a polynomial may be 0 or a Natural number; never a negative Integer.
Hi, WWGD. I hope you are doing well.
Thanks for commenting. In my definition stated above, I am interested in the singular points of ##Q_{n}(x)##, which are obtained by the zeros or solutions of ##P_{n}=0##. In as much as ##(Q_{n}(x))^{-1}=P_{n}##, then I have considered ##(Q_{n}(x))^{-1}=P_{n}=0##. Is that incorrect? Thanks again.
 
martinbn said:
Why not just write that ##J_n=n##?
Hi, martinbn, I hope you are doing well. Thanks for commenting.
It could be ##n##. However, I would prefer ##J_{n}## rather than simply ##n##.
 
V9999 said:
Hi, martinbn, I hope you are doing well. Thanks for commenting.
It could be ##n##. However, I would prefer ##J_{n}## rather than simply ##n##.
Why!? It is the maximal number of roots a polynomial of degree ##n## can have, which is ##n##.
 
  • #10
V9999 said:
Let
##Q_{n}(x)## be the inverse of an nth-degree polynomial. Precisely,

$$Q_{n}(x)=\displaystyle\frac{1}{P_{n}(x)}$$,
Your definition for ##Q_n(x)## would be the reciprocal of ##P_n(x)##, not the inverse.

For example, if ##f(x) = 2x + 3##, then ##f^{-1}(x) = \frac 1 2(x - 3)##. Note that the reciprocal of f would be ##\frac 1 {2x + 3} \ne \frac 1 2 (x - 3)##.

The operation involved in a function and its inverse is function composition. The operation involved in a function and its reciprocal is multiplication.
 
  • Like
Likes V9999 and WWGD
  • #11
Mark44 said:
Your definition for ##Q_n(x)## would be the reciprocal of ##P_n(x)##, not the inverse.

For example, if ##f(x) = 2x + 3##, then ##f^{-1}(x) = \frac 1 2(x - 3)##. Note that the reciprocal of f would be ##\frac 1 {2x + 3} \ne \frac 1 2 (x - 3)##.

The operation involved in a function and its inverse is function composition. The operation involved in a function and its reciprocal is multiplication.
And 0s would be singularities. Op's framing seemed a bit confusing to me.
 
  • Like
Likes V9999 and Mark44
  • #12
Mark44 said:
Your definition for ##Q_n(x)## would be the reciprocal of ##P_n(x)##, not the inverse.

For example, if ##f(x) = 2x + 3##, then ##f^{-1}(x) = \frac 1 2(x - 3)##. Note that the reciprocal of f would be ##\frac 1 {2x + 3} \ne \frac 1 2 (x - 3)##.

The operation involved in a function and its inverse is function composition. The operation involved in a function and its reciprocal is multiplication.
Hello, Mark44. I hope you are doing well!
Thanks a lot for your insightful comment. That is exactly what I was thinking. However, how may I mathematically define the "reciprocal" of ##Q_n(x)##?
That is to say, is there a specific notation to define the reciprocal of ##Q_n(x)##? Thanks in advance, and my apologies for the delay.
 
  • #13
V9999 said:
Hello, Mark44. I hope you are doing well!
Thanks a lot for your insightful comment. That is exactly what I was thinking. However, how may I mathematically define the "reciprocal" of ##Q_n(x)##?
That is to say, is there a specific notation to define the reciprocal of ##Q_n(x)##?
How about ##\frac 1 {Q_n(x)}##?
 
  • #14
It's not clear to me what you are trying to do.
In post #1 you gave this definition:
##J_{n}=\text{Sup}\{\pi(Q^{-1}_{n}(x)=0):\partial Q_{n}^{-1}(x) \leq n\}##

But since, after some time, we learned that ##Q_n(x)## is just the reciprocal (or multiplicative inverse) of ##P_n(x)##, the above could be rewritten as ##J_{n}=\text{Sup}\{\pi(P_{n}(x)=0):\partial (P_{n}(x) \leq n\}##

Some comments:
1. It's good that you defined what you mean by ##\partial(P_n(x))##, because I haven't seen that symbol used to mean "number of solutions of". In any case, as pointed out by @martinbn in post #9, a polynomial of degree n has n solutions.
2. What does the notation ##\pi(P_n(x))## mean?
3. What are you trying to do? In your summary, you 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.
As noted above, a polynomial of degree n has n roots, some of which might be complex. It seems to me that you are trying to obsccure a very simple idea with complicated set-theoretic notation.

Please clarify what you are trying to do, and why you want to do this.
 
Last edited:
Back
Top