- #1

V9999

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- 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.

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

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.