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When introducing the techniques of differentiation, polynomials come up all the time as great examples to familiarize students with the "power rule" and the linearity of differentiation.

A common extension is to then work with expressions involving terms with negative degree, like $3x^2 - 2x + 5 - \frac{1}{x} + \frac{3}{x^2}$.

I was wondering if there was a canonical way to refer to such expressions.

Considerations so far

The closest terminology I have come across in my own study is the Laurent series, but I find it non-ideal because of

  • The association with complex numbers
  • The infinite nature of the series (though this could be solved by invoking a "formal Laurent Series")

Other names that I have considered include the rational function, but I feel that this refers to an expression more akin to $\frac{P(x)}{Q(x)}$ which is more relevent in the discussion of the quotient rule.

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    $\begingroup$ In classes I taught I used to call these "polynomial type functions" (probably since the late 1980s, if not earlier), and for me they included fractional and even irrational exponents -- thus something in the ${\mathbb R}$-linear span (vector space sense) of $\{x^r:\; r \in \mathbb R \}.$ However, I don't know of a standard name (or even a semi-standard name) for this, or for the smaller collection of functions you're asking about. $\endgroup$
    – Dave L Renfro
    Commented Feb 3, 2023 at 12:24
  • $\begingroup$ I think "rational function" is it. There's also "proper rational function" for the case of $\operatorname{deg}(P)<\operatorname{deg}(Q)$. $\endgroup$
    – user170231
    Commented Feb 3, 2023 at 23:31
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    $\begingroup$ @user170231 Please do not use comments to post answers. $\endgroup$
    – Federico Poloni
    Commented Feb 4, 2023 at 11:48
  • $\begingroup$ I encountered the term "signal" in my linear algebra class, but I think it just refers to a bidirectionally infinite sequence, which is closely related but not quite the same. $\endgroup$
    – Solomon Ucko
    Commented Feb 4, 2023 at 23:25
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    $\begingroup$ Note that "rational functions" is more general, but not incorrect. That is, your functions are a particular type of rational functions: they can be written as P(x) / x^n for some n, which is a particular case of P(x) / Q(x). $\endgroup$
    – Stef
    Commented Feb 5, 2023 at 11:41

7 Answers 7

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There is also the term "Laurent series", where we allow an infinite number of terms... $$ \dots +5x^{-3} + 2x^{-2} + 3x^{-1} + 2 + 4x - 7x^2+\dots $$ So perhaps yours is a "finite Laurent series" or maybe "Laurent polynomial".

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    $\begingroup$ I think Laurent polynomial is pretty standard: en.wikipedia.org/wiki/Laurent_polynomial $\endgroup$
    – Eike Schulte
    Commented Feb 3, 2023 at 14:29
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    $\begingroup$ Laurent polynomial is absolutely the terminology used for this in research math. $\endgroup$
    – David E Speyer
    Commented Feb 3, 2023 at 15:27
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    $\begingroup$ I think this answer might be improved if it was modified to say "If there was a commonly used term for these, it would be "Laurent polynomials". The term "Laurent polynomials" is not widely used." (at least that's my impression.) $\endgroup$
    – JonathanZ
    Commented Feb 4, 2023 at 17:45
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    $\begingroup$ @JonathanZsupportsMonicaC, in my math world, "Laurent polynomial" is completely standard... $\endgroup$
    – paul garrett
    Commented Feb 4, 2023 at 19:42
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You can call it a "polynomial in $x$ and $x^{-1}$" or a "polynomial function of $x$ and $x^{-1}$". The idea is that you are taking a two variable polynomial $p(x,y)$ and then substituting $y=x^{-1}$.

Similar terminology is also useful so that you can say things like "this technique allows me to integrate any rational function of sine and cosine".

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  • $\begingroup$ Note that this implicitly defines an equivalence relation on the space of two-variable polynomials, because if P(x,y) = xy then P(x, x^-1) = 1, so that all the mixed terms cancel out and every polynomial P(x, y) is equivalent to a polynomial of the form Q(x) + R(y). $\endgroup$
    – Stef
    Commented Feb 5, 2023 at 11:51
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In comments, Eike Schulte and paul garrett mentioned that the standard term seems to be "Laurent polynomials."

Gerald Edgar's answer also mentions this term in passing, but I figured that it may be worthwhile to have an answer that features it front-and-center.

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The question is about what to call these expressions/functions when teaching the first basic properties of derivatives. All of the names proposed in previous answers would just make this harder for the student. They would be better off with this collection of expressions/functions having no name.

Dave Renfro's comment seemed useful to me, so I'm bumping it up to an answer, with some added thoughts of my own. He wrote: "In classes I taught I used to call these "polynomial type functions" (probably since the late 1980s, if not earlier), and for me they included fractional and even irrational exponents..."

The textbooks typically include negative and fractional powers, along with the positive powers (even though the extension of the power rule to these powers has not yet been proved). I call them "polynomial-like functions", very similar to Dave's "polynomial type functions". Thus I have a name for what is included, but I'm not asking my students to learn some new (not particularly useful) vocabulary.

I recommend that we keep it simple here.

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    $\begingroup$ I really wish I could accept two answers here: one for the forward/student facing terminology, which I whole-heartedly agree with your well thought out answer (many thanks!), and one for the more backend terminology (eg for a code library I'm also working on), both of which led me to ask the qn! $\endgroup$
    – Kelvin Soh
    Commented Feb 6, 2023 at 13:10
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I would also like to suggest posynomials $$f(x_1,\cdots,x_n)=\sum_{k=1}^Kc_kx_1^{a_{1,k}}\cdots x_n^{a_{n,k}}.$$ These generalise the idea to a multivariate setting with interaction terms, and the exponents can be any real numbers.

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    $\begingroup$ I was thinking "posynomials" myself, but those have an important further restriction we don't want here: the coefficients should be positive. $3x^2 - 2x + 5 - \frac{1}{x} + \frac{3}{x^2}$ is not a posynomial. $\endgroup$
    – Misha Lavrov
    Commented Feb 4, 2023 at 4:05
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    $\begingroup$ (Wikipedia suggests that the term we want is a signomial - which I've never encountered personally.) $\endgroup$
    – Misha Lavrov
    Commented Feb 4, 2023 at 4:59
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Rational function fits the description, as we can always combine the terms into a single ratio of polynomials.

$$\underbrace{\overbrace{3x^2 - 2x + 5}^{\rm polynomial} + \overbrace{ - \frac{1}{x} + \frac{3}{x^2}}^{\text{proper rational}}}_{\rm rational}$$

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    $\begingroup$ But there are oodles of rational functions that CAN'T be put into this "Laurent polynomial" form, so "rational function" is not a $\ldots$ way to refer to such expressions. (Of the several million words I've posted on the internet since the late 1990s, I don't think I've ever used the word "oodles". Now I have $\ldots)$ $\endgroup$
    – Dave L Renfro
    Commented Feb 4, 2023 at 21:45
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Another option is to use multinomial or algebraic multinomial. However, it should be noted that this term is sometimes used as an AKA of polynomial.

References:

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    $\begingroup$ Multinomials would inlcude expressions like x² + y with more than one variable. $\endgroup$
    – James K
    Commented Feb 5, 2023 at 11:14
  • $\begingroup$ @JamesK: Agreed. But does that make the use of "multinomial" incorrect for the expression given in the question? $\endgroup$
    – Peter Grill
    Commented Feb 6, 2023 at 17:30
  • $\begingroup$ It doesn't answer the question "Is there a canonical name for a polynomial-like expression allowing for negative powers" Since it includes many things that aren't Laurant polynomials. $\endgroup$
    – James K
    Commented Feb 6, 2023 at 23:27
  • $\begingroup$ Ok, thanks for the clarrification. So, Laurent Polynomials can't be multivariable? $\endgroup$
    – Peter Grill
    Commented Feb 7, 2023 at 9:27

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