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Famous Theorems of Mathematics/π is transcendental/Monomial ordering

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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