Famous Theorems of Mathematics/π is transcendental/Monomial ordering


Definition 1

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Let   be an N-tuple. Let us define:

 

For functions   let us define:

 

This abbreviated notation will be of great use to us in the following pages and in the proof.

Definition 2

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Let   be a field. Then   is the polynomial space in variables   with coefficients in  .

A monomial is a polynomial of the form  , such that   and  .

Definition 3

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Let  , and let   be an exponent vector. Let us define:

  •  
  •  
  • The degree of a non-zero polynomial is equal to the maximum of the degrees of its compsing monomials.

Properties

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Monomial multiplication maintains exponent vector addition:

 

Definition 4

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Let   be monomials.

We say that   is of lower order than   (and denote it by  ) if there exists an index   such that

 

In other words, the vectors   have a lexicographic ordering.

In a polynomial  , the monomial of maximal order is called the leading monomial, and is denoted by  .

Example

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Let   be polynomials. Then  .

Proof

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Let   be monomials, with  .

1. Let us assume that  . We will show that   for all  .
By definition, there exists an index   such that

 

2. Let us assume also that  . We will show that  .
By definition, there exist indexes   such that respectively

 

hence:

 

 

Definition 5

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Let   be a polynomial. Let us define:

 

Meaning, the set of all monic monomials of degree   which are of lower order than  .

Example

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Monomial ordering