MF60: What exactly is a polynomial?

Polynomials are fundamental objects in algebra, but unfortunately most accounts of them skimp on giving a proper definition. Here we base polynomials on the more basic objects of polynumbers.

We introduce the particular positive polynumber alpha, and show that any polynumber can be written as a linear combination of powers of alpha. Then we define a positive polynomial to be a positive polynumber written in this standard alpha form.

Category:

Science & Technology

Tags:

• Mathematics
• foundations
• algebra
• polynomials
• polynumbers
• arithmetic
• alpha
• Wildberger

[lopecho1]

If I might ask, just for clarification. You said α^7 + 2α^4 was not a positive polynumber because the exponents weren't decreasing from left to right, but wouldn't α^7 + 2α^4 == 2α^4 + α^7 by the commutative property of addition? Pardon my question if I missed something, I just was a little confused.

[njwildberger]

@lopecho1 We are establishing a convention for when a certain expression is allowed to be called a positive polynumber. I want to draw attention to the subtle distinction between an expression and what it can be evaluated to. I will be talking about this more: here is a simpler example: is 3+4 a natural number? or is it rather an arithmetical expression that can be evaluated to a natural number? I prefer the latter position!

[newtonparticle]

Just out of interest, is there an analog between the operations you have described and any vector-vector operators from linear algebra?

[njwildberger]

Hi newtonparticle,

Its a reasonable question, but no such analog comes to mind.

[pmccarthy001]

I'm confused, perhaps I missed something... but, isn't the variable of a polynomial, like x in x^2 + 3x + 5, usually taken to be some arbitrary value over a particular field or ring, like the real, or complex numbers? If I'm following correctly, it sounds like your definition of 'polynomial' is putting a restriction on that...? Am I misunderstanding what you're saying?

[lopecho1]

@njwildberger Okay, I see your point. Thanks for the clarification.