Number of monomials of degree at most d

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In summary, the number of monomials in a finite field of degree at most d can be shown using the formula n+d choose n, where n is the number of coefficients and d is the degree of the field. This formula can be verified by trying some simple cases, such as the field of two elements, and noticing a pattern. The paper being referenced states that a finite field F[x_1, ..., x_n] of degree at most d has (d+1)^n monomials.
  • #1
mathstime
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How does one show that the number of monomials in a finite field of degree at most d is

n+d choose n?? (sorry, I don't know how to write this in maths language)

In theory I know how to do this, but I am making a mistake somewhere along the line and get a different answer.
 
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  • #2
How many choices do you have for each coefficient?
How many coefficients can you choose?

It may help to try some simple cases... if you have the field of two elements, how many quadratic polynomials are there? Cubic polynomials? You should notice a pattern pretty quickly
 
  • #3
the paper I am reading says:

Finite field F[x_1, ..., x_n] of degree at most d
 

What is the definition of "number of monomials of degree at most d"?

The number of monomials of degree at most d refers to the total number of terms in a polynomial expression where the degree of each term is equal to or less than d.

How do you calculate the number of monomials of degree at most d?

To calculate the number of monomials of degree at most d, you can use the formula (d+1)^n, where n is the number of variables in the polynomial expression. This formula assumes that all variables have a non-negative exponent.

Can the number of monomials of degree at most d be negative?

No, the number of monomials of degree at most d cannot be negative. It is always a positive integer.

What is the significance of the number of monomials of degree at most d?

The number of monomials of degree at most d is significant because it represents the number of distinct terms in a polynomial expression of a certain degree. It can also help in determining the complexity of a polynomial expression.

How does the number of monomials of degree at most d change with increasing d?

The number of monomials of degree at most d increases with increasing d. The rate of increase depends on the number of variables in the polynomial expression. As d increases, the number of monomials increases at a faster rate.

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