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

In summary, the reciprocal of an nth-degree polynomial is defined as the function that returns the value 1 divided by the value of the nth-degree polynomial at a given x-value.
  • #1
V9999
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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.
 
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  • #2
Why are we even talking about Q instead of P?
 
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  • #3
Why not just write that ##J_n=n##?
 
  • #4
##\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.
 
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  • #5
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.
 
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  • #6
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.
 
  • #7
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.
 
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  • #8
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##.
 
  • #9
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.
 
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  • #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.
 
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  • #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)}##?
 
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  • #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.
 
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What is set theory?

Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects. It is used to formalize the concept of mathematical sets and their properties.

How can set theory be used to define the number of solutions of polynomials?

Set theory can be used to define the number of solutions of polynomials by representing the solutions as elements of a set. The number of elements in this set would then correspond to the number of solutions of the polynomial.

Is set theory the only way to define the number of solutions of polynomials?

No, set theory is not the only way to define the number of solutions of polynomials. Other mathematical concepts, such as algebraic geometry and complex analysis, can also be used to define the number of solutions.

Can set theory be used for all types of polynomials?

Yes, set theory can be used to define the number of solutions for all types of polynomials, including linear, quadratic, cubic, and higher degree polynomials.

Are there any limitations to using set theory to define the number of solutions of polynomials?

While set theory can be a useful tool for defining the number of solutions of polynomials, it may not always provide a complete understanding of the solutions. In some cases, other mathematical concepts may be needed to fully describe the solutions of a polynomial.

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