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Let $F$ be a field. What is the best notation (in an undergraduate or graduate abstract algebra class) for a generic element of the univariate polynomial ring $F[x]$?

The most common notation seems to be $f(x)$. It seems strongly to me that $f$ (or any other single letter) is to be preferred: it emphasizes that elements of $F[x]$ are well-defined objects, without reference to plugging in anything for $x$, and that if you see $f(a)$, this always indicates the evaluation homomorphism obtained by replacing $x$ with $a$.

There is no logical conflict here, because the tautological evaluation homomorphism $F[x] \rightarrow F[x]$ sending $x$ to $x$ is just the identity. Nevertheless, I cannot help but feel that writing $f(x)$ sends the wrong message.

Nevertheless, judging from a skimming of the algebra textbooks on my shelf, I seem to be in the minority. Is there reason to prefer writing $f(x)$?

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    $\begingroup$ It depends on how you are going to use it. If you are going to take the formal derivative, to suggest an analogy with the derivative operation in calculus, f(x) might be a good choice. Since it will be a polynomial which you are likely to do algebraic things with, I think p(x) or P(x) might be preferable. If you want to stress the abstract nature of the object, call it Fred. In fact, certain programming environments encourage aliases for a certain object to enable viewing the object in a different light. Gerhard "Where Will Fred Go Today?" Paseman, 2015.03.02 $\endgroup$
    – Gerhard Paseman
    Commented Mar 2, 2015 at 17:56
  • $\begingroup$ In addition to aliases for an object, there is also the issue of representation. In executing the evaluation homomorphism, it is useful to represent Fred as a certain sum of monomials, which makes clear how to perform the evaluation using the member a from F. You may want to make the distinction between representing and renaming clear, to help students understand how to work with things like Fred. Gerhard "What Will Fred Do Today?" Paseman, 2015.03.02 $\endgroup$
    – Gerhard Paseman
    Commented Mar 2, 2015 at 18:00

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A couple of practical points to consider:

  1. If one does not always write the variable it can be more tricky to avoid confusion between substitution and multiplication. Consider:

    Let $f=5x^2 +7$. Further, let $g = f(x^2 +1)$.

    What is meant here, $5(x^2+1)^2 +7$ or rather $(5x^2+7)(x^2+1)$?

    By contrast

    Let $f(x)=5x^2 +7$. Further, let $g(x)= f(x^2 +1)$.

    and

    Let $f(x)=5x^2 +7$. Further, let $g(x)= f(x)(x^2 +1)$.

    seem both clear.

  2. Another issue arises when there are parameters involved: "the polynomial $f=x^2 + ax+ a^2$" is somewhat unclear. By contrast "the polynomials $f(x)= x^2 + ax+ a^2$ and $f(a)=x^2 + ax+ a^2$ and $f(x,a)=x^2 + ax+ a^2$" all seem clearer.

  3. Similar to the previous point it can be handy to be able to highlight in the multi-variable case when some polynomial does not depend on all variables, say like: "We have a factorization $f(x,y)= g(x)h(y)$."

That being said I do agree with the point that there is merit in pointing out that there is no actual need to write $f(x)$. And, one could avoid the issues mentioned above in other ways. Still there is also some downside, and as you asked about this I mainly talk about this.

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    $\begingroup$ I would disagree with your use of notation. Writing $f=5x^2+7$ isn't right. Instead, $f$ is the function given by $f(x)=5x^2+7$, or $f$ is the polynomial $5x^2+7$. $\endgroup$
    – Jessica B
    Commented Mar 3, 2015 at 8:15
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    $\begingroup$ I am not sure in which sense it isn't right. The string $5x^2 +7$ signifies a unique element of, say, the ring $\mathbb{Q}[x]$. I think I can decide to call this element $f$ by writing $f= 5x^2 + 7$. Just as I can write let $c = \sqrt{5}+ 19/3$ to give a name to a specific real number signified by some other string. $\endgroup$
    – quid
    Commented Mar 3, 2015 at 9:10
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    $\begingroup$ @JessicaB These are purely algebraic objects in the polynomial ring in x which in independent from the base ring. They do not operate on an underlying set in this sense. $\endgroup$
    – Chris C
    Commented Mar 3, 2015 at 13:59
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    $\begingroup$ A thoughtful response. Thank you! $\endgroup$
    – Frank Thorne
    Commented Mar 3, 2015 at 16:28
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    $\begingroup$ You are welcome! I wondered about this question, too, from time to time. I feel writing this down helped me clarify my own thinking on the subject. $\endgroup$
    – quid
    Commented Mar 3, 2015 at 16:30
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I would strongly present notation that is consistant with whatever book you are using. I know the most popular books use the $f(x)\in F[x]$ (Dummit and Foote, Lang), while others use just $f \in F[x]$ (Atiyah and Macdonald). Either way, they both represent the same polynomial $a_0 + a_1 x + ... + a_n x^n \in F[x]$.

I would strongly indicate that notation for the variable (or other object) is defined by the $F[x]$, and not the representation of the elements. You might even indicate $f = f(x) \in F[x]$ so that they can see the equivalence. Students should be warned that $f(c) \in F$ for $c \in F$.

Personally, I like the notation $f(x) \in F[x]$. It highlights the evaluation map $Eval_a :F[x] \to F$ by $Eval_a (f(x))=f(a)$ instead of the map $x \mapsto a$, but for a sufficiently advanced class it shouldn't matter. The definition element-wise might be easier for beginning students.

But as @quid pointed out, if you mix notations, ambiguity arises. So stick with your book's notations and highlight them early.

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    $\begingroup$ Minor nitpick: one should be careful about referring to polynomials as functions (especially in an introductory abstract algebra course). There's a natural homomorphism from $F[x]$ to the ring of functions $F \to F$ with pointwise addition and multiplication, but this homomorphism fails to be injective if $F$ is finite. $\endgroup$
    – Daniel Hast
    Commented Mar 2, 2015 at 19:57
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    $\begingroup$ @DanielHast Duly noted and adjusted. Slipped out due to habit. $\endgroup$
    – Chris C
    Commented Mar 2, 2015 at 20:19
  • $\begingroup$ +1 for follow the textbook. Especially for an undergraduate class. $\endgroup$
    – Gerald Edgar
    Commented Mar 2, 2015 at 21:43
  • $\begingroup$ Good point, thank you. $\endgroup$
    – Frank Thorne
    Commented Mar 3, 2015 at 16:28
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Essentially following up on Daniel Hast's comment (Notation for an element in a polynomial ring): I think that the usual reason to distinguish between $f$ and $f(x)$ is to avoid a confusion between a function and its value; but, in this case, there is a further confusion to avoid, which is that between a polynomial and 'its function' (or rather its family of functions, since a polynomial in $R[x]$ gives a function $A \to A$ for every $R$-algebra $A$). I think that this argues for the notation $f(x)$, because it emphasises (to those 'in the know') that $f(x)$ is not a function.

I guess the real question is: what is your name for the obvious degree-$1$ polynomial? It seems reasonable to call it $x$ (else what does the $x$ in the notation $R[x]$ mean?); but then, as quid points out at Notation for an element in a polynomial ring, your element really and truly is (say) $5x^2 + 7$, not "the function whose value at $x$ is $5x^2 + 7$".

Overall, I think that, as you point out yourself, there is nothing logically wrong, by even the pickiest standards, with the notation $f(x)$, whereas there are solid reasons (well elucidated by quid at https://matheducators.stackexchange.com/a/7514/2070) not to use $f$; so, to the extent that the notation is yours (rather than forced on you by the need for consistency), I would advise going with the former.

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One quirky feature of polynomial rings is that if $f \in F[x]$, $a$ is an element of an $F$-algebra, and the notation $f(a)$ means the result you get from evaluating $f$ at $a$ (i.e. applying the relevant evaluation homomorphism), then $f = f(x)$ is actually a literally true statement.

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Why not indulge in a bit of redundancy and write for instance

$x \xrightarrow{\hspace{5mm} f\hspace{5mm}} f(x) = -3x^2 +2x -21$

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