Proving (n-1)th Degree Polynomial Representation of n Points

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Discussion Overview

The discussion centers on proving that a polynomial of degree at most (n-1) can represent n distinct points in a given space. Participants explore the implications of polynomial degree requirements based on the number of points and the context of representation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that a maximum of a (n-1)th degree polynomial is needed to represent n points, citing the need for maximum/minimum points.
  • Another participant questions the meaning of "represent" and the use of "maximum," asking for clarification on the dimensionality of the space involved.
  • A participant explains that for 1 point, a 0th degree polynomial suffices, for 2 points a 1st degree polynomial is needed, and for 3 points a quadratic is necessary.
  • Another participant asserts that for n points with distinct x-coordinates, a degree (n-1) polynomial exists that passes through them, emphasizing the uniqueness of this polynomial.
  • One participant references the Lagrange Interpolation method as a straightforward approach to demonstrate the polynomial representation.
  • A later reply reiterates the effectiveness of the Lagrange Interpolation method for any n points, though it does not directly address the proof of needing a specific degree.
  • Another participant mentions the Vandermonde determinant as a proof of the linear independence of the equations formed by substituting the n points.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of the maximum degree polynomial and the interpretation of representation. There is no consensus on the best approach to proving the claim, and multiple perspectives on the topic remain unresolved.

Contextual Notes

Participants have not fully defined the conditions under which the polynomial representation holds, such as the requirement for distinct x-coordinates or the dimensionality of the space. The discussion includes various assumptions that have not been explicitly stated.

Alkatran
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Let's say I wanted to prove that, given n points, it takes a maximum of a (n-1)th degree polynomial to represent them all. How would I do it? My instinct is to just say because you need a max of (n-1) max/mins ...
 
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what does represent mean in this context? and are you sure you mean "maximum"

what degree one polynimial "represents" the two points 0 and 1. and point in what space? R, R^2, R^3...?
 
I mean that, given 1 point that your equation must touch, you need a 0th degree equation. Given 2 you need a 1st, etc...

For example, if you are given a set of points with 2 elements:
(a,b), (c,d)
You need a 1st degree equation, or line.
y = mx + e
The correct value of m and e will hit both points.

Similarly, if you have 3 points, you need a quadratic.
 
In that case, given n points in the plane with distinct x coords, there exists a degree n-1 polynomial passing through them, since a degree n-1 poly has n coefficients and therefore you have a system of n linearly independent equations in n unknowns to solve.

you don't mean maximum at all since given n points then there is a polynomial of degree r=>n-1 passing through those points (again with distinct x values) which is unique when r=n-1.
 
Alkatran,
If u go through Lagrange Interpolation method, u would see how lagrange came up with an extremely simple way to do it!
 
TenaliRaman said:
Alkatran,
If u go through Lagrange Interpolation method, u would see how lagrange came up with an extremely simple way to do it!

I'm aware of how to solve the problem. My question was how do I prove that I will never need a 5th degree equation for 5 points?
 
Lagrange Interpolation Method works for any given n points.
Hence Proved!
 
The proof that the equations formed by substituting in the n points are linearly independent is called the vandermonde determinant.
 

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