In a polynomial of $n$ degree, what numbers can fill the $n$?

Question

Until now, I've seen that the $n$ could be filled with the set $\mathbb{N}_0$ and $-\infty$ but I still didn't see mentions on other sets of numbers. As I thought that having 0 and $-\infty$ as degrees of a polynomial were unusual, I started to think if it would be possible for other numbers to also fill the gap.

Answer 1

So far you are talking about polynomials. I assume the coefficients are in some field $k$ so you would denote them as $k[x]$. Here the degree is in $\mathbb N_0\cup\{-\infty\}$.

Common generalisations are:

1. The ring $k[x,x^{-1}]$. So these are polynomials in the two variables $x$ and $x^{-1}$ with the relation $x\cdot x^{-1}=1$. Then we can define the degree similar to before as the highest non-vanishing power of $x$. Thus the degree will be in $\mathbb Z\cup\{-\infty\}$. Another name for these objects is Laurent polynomials.

2. The ring of formal power series $k[[x]]$. Here you allow also the degree $\infty$ for an formal infinite sum $\sum_{i=0}^\infty a_ix^i$. Degrees are in $\mathbb N_0\cup\{\pm\infty\}$

3. The ring of Laurent series. This is a combination of 1. and 2. Polynomials may include negative powers and (possibly infinitely many) positive powers of $x$. Note that we only allow finitely many negative powers since we can't porperly define multiplication otherwise. In the non-algebraic world people sometimes allow infinitely many negative powers as well. The degree however will be in $\mathbb Z\cup\{\pm\infty\}$

4. The field of fractions of the polynomial ring $k(x)$. Here an element is a fraction $\frac{P}{Q}$ of two polynomials, where $Q\neq 0$. Addition and multiplication are defined similar as the same operations for ordinary fractions. The degree can be defined as $deg(P)-deg(Q)$. Check that the properties of the degree wrt to the operations are still in place. Also note that $k(x)\neq k[[x]]$, although we also have degrees in $\mathbb Z\cup\{-\infty\}$.

Answer 2

The reason for this unusual $-\infty$ is the following:

Consider a non-zero polynomial in $x$. Then its degree is simply the highest power of $x$ which has a non-zero ceofficient. Now it turns out that if you multiply a polynomial of degree $m$ with a polynomial of degree $n$, you get a polynomial of degree $m+n$ (simply because $x^m\cdot x^n=x^{m+n}$). Moreover if we add a poynomial of degree $m$ and a polynomial of degree $n>m$ then the sum has degree $m$.

So far, so good. But now we have a problem: The zero polynomial clearly also is a polynomial, but it does not have a highest non-zero coefficient (because simply all coefficients are zero). Therefore the rule above doesn't tell us which degree we should assign to it. However we can look at the properties of the zero polynomial to figure out a good choice.

First, if we multiply the zero polynomial with any other polynomial, the result is, of course, the zero polynomial. That is, if $d$ is the degree of the zero polynomial, we want to have $d+n=d$ for all $n\in\mathbb N_0$. However there's no number which fulfils this identity. But if we look at the "numbers" $\infty$ and $-\infty$, we find that they have, according to common convention, exactly this property. Therefore it makes sense to use either $d=-\infty$ or $d=\infty$.

To determine which of them to use, we look at the second property, adding polynomials of different degree, and getting a polynomial of the larger degree. Now if we add the zero polynomial to a polynomial of degree $n$, we of course get that other polynomial, with degree $n$. Therefore we want to have $n>d$, which is the case for $d=-\infty$ but not for $d=\infty$. Therefore we assign, by convention, the degree $-\infty$ to the zero polynomial.

As for other values which could be meaningful, I could imagine to look at the space of all functions defined by their power series. In that case, the functions which are power series, but not polynomials, could be given the degree $\infty$.

Another obvious extension of polynomials would be to allow negative powers (such functions would, of course, not be defined at $0$). In that case, a generalized polynomial could have a negative integer degree.

Finally, if looking at functions which are sums of arbitrary (also non-integer) powers, such a generalized polynomial would have real degree.

However you should note that in all those cases, the functions are no longer polynomials, but a generalization which include the polynomials. For polynomials, the only degrees possible are from $\mathbb N_0\cup\{-\infty\}$.

Answer 3

The degree of a polynomial can only take the values that you've specified. For that, let's revisit the definition of a polynomial. Personally, I was taught that a polynomial (in one variable) is an algebraic expression which can be written in the form $$a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$$ where $n$ is a non-negative integer. From this definition, as the degree is the same as $n$, therefore, the degree is also a non-negative integer (I'll not go into the degree of a zero polynomial, which is best left undefined).

On the other hand, Wikipedia says that

A polynomial is an expression of finite length constructed from variables (also called indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

Clearly, there is no non-integer expenentiation, nor any division by a variable, so again it is clear that the degree is a non-negative integer.