The Degree of the Zero Polynomial: Why is it Defined as -∞?

[marellasunny]

I understand that mathematicians have had to define the number '0' also as a polynomial because it acts as the additive identity for the additive group of poly's.What I do not understand is why they define the degree of the zero polynomial as [ tex ]-\infty[ /tex ].

An explanation on planetMath wasn't that helpful,at the end they point-out to refer to the extended real numbers(don't they mean 'projectively extended real numbers??)

 

[Mute]

Well, I guess it's similar to how one sometimes regards zero as both a real and an imaginary number because you can write 0 = 0 + i0. Similarly, you can write

0 = 0 + 0x + 0x² + 0x³ + ...

i.e., you can write '0' as an infinite degree polynomial with all coefficients zero.

(There may be a more rigorous reason, but that's an intuitive one).

 

[marellasunny]

Thanks!Could you explain what extended real numbers have got to do with this?
But,polynomials always have non-negative degrees.
deg[P(x)]=+n
Why would mathematicians define a polynomial with a negative degree?

 

[eumyang]

Huh, I'm looking at a precalculus textbook (Larson, 8th Ed.), and it states that the zero polynomial has no degree. Is that wrong? (Note that no degree ≠ zero degree -- a polynomial that consists of a single non-zero number has a degree of zero.)

 

[micromass]

The choice is pretty arbitrary. Sometimes it defined as having no degree, sometimes it's -1, sometimes it's $-\infty$.

A handy formula for polynomials is

$$deg(P)+deg(Q)=deg(PQ)$$

If we want this formula to hold for the zero polynomial, then we see (by taking Q=0) that

$$deg(P)+deg(0)=deg(0)$$

must hold for all P. This is only satisfied with $deg(0)=-\infty$. This is the reason why they defined it this way. But again, it's pretty arbitrary.

 

[eumyang]

This is only satisfied with $deg(0)=-\infty$. This is the reason why they defined it this way. But again, it's pretty arbitrary.

I see now. Thank you.

 

[marellasunny]

The choice is pretty arbitrary. Sometimes it defined as having no degree, sometimes it's -1, sometimes it's $-\infty$.

A handy formula for polynomials is

$$deg(P)+deg(Q)=deg(PQ)$$

If we want this formula to hold for the zero polynomial, then we see (by taking Q=0) that

$$deg(P)+deg(0)=deg(0)$$

must hold for all P. This is only satisfied with $deg(0)=-\infty$. This is the reason why they defined it this way. But again, it's pretty arbitrary.

Awesome!Thanks a ton.