Or instead of talking about polynumbers, one could just use the normal set-theoretic definition: A polynomial over a ring R is a finitely supported function from ℕ₀ to R.
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?
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.
@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!
This has been flagged as spam show
Or instead of talking about polynumbers, one could just use the normal set-theoretic definition: A polynomial over a ring R is a finitely supported function from ℕ₀ to R.
inf0phreak 1 week ago
Comment removed
inf0phreak 1 week ago
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?
pmccarthy001 1 month ago
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.
lopecho1 2 months ago
@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!
njwildberger 2 months ago
@njwildberger Okay, I see your point. Thanks for the clarification.
lopecho1 2 months ago
Just out of interest, is there an analog between the operations you have described and any vector-vector operators from linear algebra?
newtonparticle 1 year ago
Hi newtonparticle,
Its a reasonable question, but no such analog comes to mind.
njwildberger 1 year ago