I What type is a polynomial function?

[iteratee]

TL;DR Summary

Is there a universal definition and purpose for considering these a distinct category?

I often encounter functions called "polynomial" in numerous fields. I don't see an obvious common trait other than that they're usually describing a real-valued continuous function. What aspects are typical or universal or distinct? What structures can be polynomial? Some sources say that polynomials may be defined as conforming to a grammar of sorts, as basically a sum of products (assuming numeric algebras), but in some contexts they're expressed in an implicit equation with no distinct features other than having an = sign buried within. I can't judge how such scrambled equations were derived, whether they're a special subset of a larger class of function, whether / when they can be uniquely mapped back to a common normalized form.

This is looking like either a frequently misused `term or perhaps overloaded with meanings making it oddly hard to research.

 

[PeroK]

https://www.mathsisfun.com/algebra/polynomials.html

 

[mfb]

Polynomial functions can be defined by a polynomial. It's that easy.
A polynomial function doesn't have to be real-valued. Every polynomial function is continuous but not every continuous function is a polynomial function.
There are many interesting theorems that only apply to polynomial functions. Wikipedia has examples.

 

[iteratee]

Polynomial functions can be defined by a polynomial.

Indeed I'm aware of the ubiquitously self-referential jargon surrounding this subject. Maybe there's a reason... It's largely why I've resorted to asking such a general question.

 

[Drakkith]

Indeed I'm aware of the ubiquitously self-referential jargon surrounding this subject. Maybe there's a reason... It's largely why I've resorted to asking such a general question.

It's not self-referential. It's a definition. A polynomial function is a type of function that is defined as being composed of a polynomial, which is a mathematical expression that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. Other types of functions aren't polynomials, such as the function $f(x) = e^x$, which is an exponential function, and both types are more generally elementary functions.

See here for a list of mathematical functions.

 

[PeroK]

Indeed I'm aware of the ubiquitously self-referential jargon surrounding this subject. Maybe there's a reason... It's largely why I've resorted to asking such a general question.

If we start with a single variable, then a polynomial (of degree $n$) is a function of the form: $$p(x) = a_0 + a_1 x + a_2 x^2 \dots + a_n x^n$$ And that's all there is to it.

 

[jbriggs444]

If we start with a single variable, then a polynomial (of degree $n$) is a function of the form: $$p(x) = a_0 + a_1 x + a_2 x^2 \dots + a_n x^n$$ And that's all there is to it.

One can dig a bit deeper and distinguish between a polynomial function and a formal polynomial.

With a polynomial function, one has a function (with a domain and a range and a mapping of elements in the domain to elements in the range) where the mapping matches a polynomial expression. One can add, subtract or multiply polynomial functions to get new polynomial functions.

With a formal polynomial, one has the algebraic field from which the coefficients are drawn and a finite array of coefficients $a_0$ through $a_n$. One can add, subtract or multiply these formal polynomials to get new formal polynomials.

It is tempting to think that the distinction is merely one of naming. However the truth is otherwise.

Consider polynomials over the finite field with two elements (one and zero). There are only 4 distinct functions over this domain: f(x) = 0, f(x) = 1, f(x) = x and f(x) = x - 1. All four are polynomial functions. However, there are infinitely many distinct formal polynomials.

 

[Stephen Tashi]

Indeed I'm aware of the ubiquitously self-referential jargon surrounding this subject. Maybe there's a reason... It's largely why I've resorted to asking such a general question.

You have a legitimate concern, but I wouldn't call the usual way of defining polynomials "self-referential".

Some sources say that polynomials may be defined as conforming to a grammar of sorts, as basically a sum of products (assuming numeric algebras)

Yes, that's the standard definition.

but in some contexts they're expressed in an implicit equation with no distinct features other than having an = sign buried within.

I'm not sure what you mean. Of course, a text can say something like "Let $f(x)$ be a polynomial function of degree 3" without writing out $f(x) = Ax^3 + Bx^2 + Cx + D$. Likewise a text may say "Let $M$ be a 3x3 matrix" without writing out the 9 entries of $M$.

The legitimate concern about defining a polynomial the standard way (as a symbolic expression that obeys certain syntax) is that this type of definition can fail.

For example, many USA secondary school textbooks define raising a number to a rational power by saying "$x^\frac{n}{m}$ is defined to be the $n$th power of the $m$th root of $x$ when that $m$th root exists in the real number system.

By this definition $(-1)^\frac{1}{3} = -1$ but $(-1)^\frac{2}{6}$ does not exist. So the definition does not define how to raise a number to a rational power. It only defines how to raise a number to a rational power when that power is denoted in a certain way . Using that definition we cannot conclude that if $a$ and $b$ are equal rational numbers then $x^a = x^b$

So, technically, the definition of polynomial as a function that can be written in a certain symbolic way should be accompanied by proofs that the notation actually defines a unique function and that polynomials denoted by inequivalent symbolic expressions define different families of functions. Also, if we wish to assert the function $\sin(x)$ is not a polynomial, we should prove this statement, not assume it on the basis that the function is denoted in a particular way.

Polynomials are usually introduced to students at an early stage of their education in algebra. At that stage they tend to assume that notation is unambiguous. Introductory texts don't deal with the technical questions involved in showing definitions based on notation are actually proper mathematical definitions. In the case of defining polynomials in terms of notation, the definition does succeed, but only an advanced text would offer proofs of this.