Polynomial degree (order)

[ajb]

A (formal) polynomial

 

[math]P(x) = a_{n}x^{n} + \cdots + a_{2}x^{2} + a_{1}x + a_{0}[/math]

 

is said to be of degree [math]n[/math].

 

Does anyone know if there is any standard language that describes a polynomial that is strictly of the form

 

[math]P(x) = a_{n}x^{n}[/math]?

 

(One way to do this would be to introduce a filtration and then a grading, but I wanted to avoid doing so if some common language already exists.)

 

Is the word "homogeneous" use commonly for this. (I have seen it used for similar ideas).

 

Thanks

Edited September 4, 2008 by ajb
thought of "Homogeneous"

[the tree]

At a guess, I'd say that since a multiple of x to the n is easy enough to say, there isn't a special term for it.

 

Or, if for a given polynomial I only wanted to refer to one term I'd refer to it as the 'the highest term' or 'the term of x to the n' (the latter makes no sense I realise, but it's not the type of thing that would ever leave my notes).

[ajb]

Cheers.

 

It is not quite polynomials like that I am interested in, but I wondered if there was some language already in use out there for what I want.

 

Done carefully, we have a filtration. That is a polynomial of degree n has [math]x^{n}[/math] terms and lower.

 

I need to use this filtration to get a grading, that is the polynomial algebra decomposes as [math]P(x) = \bigoplus_{n}P^{n}(x)[/math]. (I hope you understand what I am saying).

 

I want to say a polynomial P has homogeneous degree [math]n[/math] if and only if P [math]\in P^{n}(x)[/math]. Which comes from graded algebra.

 

The standard use of degree would also include lower power terms, i.e. it gives a filtration.

[Dr. Jekyll]

First I thought it was monomial, but that is without any coefficient. It seems as a monomial [math]x^n[/math] with a coefficient is called "term."

 

http://mathworld.wolfram.com/Monomial.html

http://mathworld.wolfram.com/Term.html

[ajb]

Term is ok if I set it up carefully.

 

More generally one problem I keep encountering is that some mathematical ideas are intuitively very obvious. Now trying to make precise mathematical statements and well founded constructions can be a different thing! All part of the game.