/sci/ - Science & Math

>>11606415
It is usually the case that the degree of a polynomial is defined to be a natural number. This will then give a monoid homomorphism, as the Remilia-non pointed out, from the multiplicative monoid of non-zero polynomials to the additive monoid of natural numbers. We want this to happen, but this does not cover the zero polynomial, for the reason also mentioned by the Remilia anon, so we have to either say that its degree is negative infinity (which allows us to use the function again after this little tweak) or simply say it is undefined, as it can't be any natural number. Furthermore, if it was some negative number, we could pick a polynomial of degree greater than the absolute value of the degree of the zero polynomial, and then the degree of the zero polynomial would be smaller than the degree of the zero polynomial - quite problematic! Thus, it can't be defined to be any number, but it can be set to be negative infinity if one so wishes. What your professor means by saying it is not defined is essentially that he doesn't want it to be negative infinity. Note that this choice is also consistent with the degree function.