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This is perhaps unanswerable, or perhaps I am too algebraically ignorant to phrase it cogently, but:

Is there some identifiable reason that polynomials over $\mathbb{C}$, $\mathbb{R}$, $\mathbb{Q}$, $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$ are so pervasively useful in mathematics?

Is it because polynomials are in some sense the most natural functions defined on a field? I know that every function over a finite field $\mathbb{F}^n \to \mathbb{F}$ is a polynomial. And, by the Stone–Weierstrass theorem, every continuous function on an interval can be approximated by a polynomial. Is this the universal aspect of polynomials that "explains" their ubiquity?

Even tropical polynomials, which employ alternative addition/multiplication operations forming a semiring, are proving useful.

I'd appreciate your insights!

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$K[x]$ represents the forgetful functor from $K$-algebras to sets, according to math.stackexchange.com/questions/531073/… See also the universal property for polynomials as stated at arbourj.wordpress.com/2012/04/26/… –  Gerry Myerson Jun 13 at 1:33
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I am usually a big fan of your questions and answers on this site, but I fear that this one might be too broad even for CW. I won't downvote or vote for closure, but would instead suggest rephrasing along the lines of "what are some instances where polynomials appear unexpectedly, or happen to be surprisingly beneficial?". –  Vidit Nanda Jun 13 at 1:41
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I am a bit puzzled about why you suggest the Stone-Weierstrass theorem might explain the ubiquity of polynomials, since most uses of polynomials (outside of the functional calculus, say) have no direct link to the SW theorem. –  KConrad Jun 13 at 1:42
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@trb456: Yes the SW theorem is very important, but when you work with coefficients that are not the real or complex (or $p$-adic) numbers polynomials are still important and it's not due to anything involving the SW theorem. That is, there are many reasons to care about polynomials other than in their role as approximations to other functions. –  KConrad Jun 13 at 2:07
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This should be CW I think. –  Joël Jun 13 at 20:14

9 Answers 9

up vote 30 down vote accepted

Polynomials are, essentially by definition, precisely the operations one can write down starting from addition and multiplication. More formally, polynomials with coefficients in a commutative ring $R$ are precisely the morphisms in the Lawvere theory of commutative $R$-algebras. So in some sense caring about polynomials is equivalent to caring about commutative rings and, more generally, commutative algebras. See, for example, this blog post for some details; in particular, that blog post makes precise the assertion that polynomials are not only the most natural but in fact the only natural operations on commutative $R$-algebras.

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Of course, this is basically the same as the comment that polynomials constitute free commutative algebras over a given coefficient ring: the Lawvere theory is dual to the category of finitely generated free objects. –  Todd Trimble Jun 13 at 1:52
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"...polynomials are ... the only natural operations on ...": This is, to me, the compelling point. –  Joseph O'Rourke Jun 13 at 21:57
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Joseph, while I agree Qiaochu gave a good answer, it doesn't address your question with your particular choices of coefficient rings ($\mathbb{Q}, \mathbb{R}$, etc.). In other words, your question seems a bit more specific than what this answer answers. –  Todd Trimble Jun 14 at 4:14
    
Yeah, honestly I'm not sure this really addresses the question either, but then again the question is extremely broad. –  Qiaochu Yuan Jun 14 at 6:12
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@Tom: but you can run it forward; the yoga of Lawvere theories implies that you can define commutative algebras as precisely those things whose elements you can apply polynomials to. So everything comes full circle. –  Qiaochu Yuan Jun 15 at 16:04

I suspect the question is not answerable because utility can be subjective and rationalizing subjective notions leads to more arguments and less elucidation. However, this gives me an opportunity to mention a couple aspects of polynomials that deserve more press.

Polynomials are a generalization of number representation, replacing base 10 or base 2 by base x, while eliminating carry. This representation can be used in forming error correcting codes, in computing transforms, even in speeding up multiplication of large numbers. The uses Euler had for determining certain combinatorial results through manipulating power series could still be obtained by truncation, so they served as precursors to generating functions. There are also polynomial encodings used in various ways in mathematical logic, among them forms of Goedel numbering and representing certain arithmetical facts as solutions to certain systems of polynomial equations.

In general algebraic structures, one picks some basic operations and then composes them to get derived term operations, e.g x+ yxx. Substitute elements for some of the variables, and one gets polynomial operations of the structure, e.g. x + 3xx. The set of polynomial operations can be a dense set in a larger set of operations, and one now turns to representation or approximation of arbitrary functions by a polynomial operation or by a sequence of such. Many interesting problems of representation and approximation are studied, among them circuit optimization and stability of numerical algorithms. The fact that polynomials are finite computational units is our handle on problems that might need infinite amounts of computation to determine precisely.

I guess the answer is that mathematicians like to turn problems into nails, so that they can use polynomial hammers on them. If it works, why not?

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Nice point that "polynomials are finite computational units"---analogous to approximating continuous functions by polynomials. –  Joseph O'Rourke Jun 13 at 10:43

At risk of being overly bold, allow me to suggest:

Polynomials are useful because quadratic polynomials are useful.

If we can all agree that linear algebra is an indispensable tool in mathematics then it's hard to argue with the success of equipping vector spaces with quadratic structures - this is the starting point of nearly all of geometry and large portions of number theory. Even when we move on to higher degree polynomials or transcendental objects, quadratic structures generally appear as local approximations (how often do you go past the degree 2 term in a Taylor series?)

So the question becomes: why are quadratic polynomials useful? There seem to be two different but interacting reasons. The first is that quadratic functions of a real variable are always either convex or concave and therefore have a unique maximum or minimum. The second is that quadratic functions are intimately related to bilinear forms and therefore can be accessed using linear algebra. The combination of these two reasons seems to explain the success of quadratic algebra in analysis and geometry (e.g. Hilbert spaces, Riemannian manifolds). This is also part of the story behind their utility in number theory, though I'm not sure I've completely explained the importance of quadratic structures on finite fields. (Related issue: why is quadratic algebra over $\mathbb{F}_2$ so fundamental in the topology of manifolds?)

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"this is the starting point of nearly all of geometry" --- it would have been news to Euclid that he was equipping a vector space with a quadratic structure. –  Gerry Myerson Jun 13 at 9:44
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@GerryMyerson It would indeed have been news to him, but he would probably agree once he realized that $\mathbb{R}^2$ with the standard inner product is a model for the Euclidean axioms. I did not claim (or need to claim) that this is the historical starting point for geometry. –  Paul Siegel Jun 13 at 10:32
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I would phrase the first reason for the usefulness of quadratics as "They imply a notion of positivity". –  Dirk Jun 15 at 18:54

Polynomials can be used to extract information about finite sequences much in the same way as generating functions can be used for infinite sequences. However, it is good to note that generating functions are not always more suitable for such purposes than polynomials; polynomials allow more operations and convergence issues can be neglected. In fact, one does even need the underlying field to have any topological structure.

Let me give some examples where polynomials are more appropriate than generating functions. For example, the combinatorial nullstellensatz is a useful tool in combinatorics, particularly in additive problems, and its formulation is based on polynomials over a field. Another important example is that one obtains better results in the Hardy-Littlewood circle method by replacing infinite Fourier series with trigonometric polynomials. Third exmple is that manipulating suitable polynomials in $\mathbb{Z}/p\mathbb{Z}$ gives some interesting reults in number thory, such as Wolstenholme's theorem. Fourth example is that irrationality and transcendence proofs are usually based on considering suitable polynomials (for example, in the case of $\pi$).

In addition, polynomials sometimes serve as ''generalized integers'' in number theoretical contexts. Many theorems are easier to prove for them (such as Fermat's last theorem for non-constant polynomials), and they can be used to conjecture results about integers; this is how abc conjecture was found, for instance.

On the other hand, in complex analysis, polynomials are basic examples instead of generalizations. There are numerous results in complex analysis that are easy for polynomials but generalize in an interesting way to analytic functions, and on the other hand, there are also many results where the polynomials are the only exceptions due to their slow growth or rigidness. The fact that polynomials are determined by their values in finitely many points is indeed another reason for their usefulness in analysis and other branches as well.

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I like the "generalized integers" point! –  Joseph O'Rourke Jun 13 at 20:44

Polynomials turn the idea of performing arithmetic operations (a dynamic, procedural notion: performing operations) into a static object of mathematics that can be considered in its own right (a static notion: a list of instructions, or an element of a ring). This can be seen as an early example of reifiction/object-ification in mathematics, which has traditionally been quite useful, viz. the definition of algorithm, the definition of ring, the definition of category and functor, the definition of cardinal+ordinal, etc. These all take what was previously a rough or vague or intuitive or unstated or procedural/dynamic mathematical idea and turn it into a single mathematical object that can then be studied as such. That is, one can study the object in its own right, its relation with other objects of the same or different types, etc. (In particular, turning processes into objects has been quite useful, but this is but one kind of a more general phenomenon.)

Now, why are polynomials specifically so incredibly and widely useful? Perhaps because the thing they are reifying is so basic, namely the arithmetic operations.

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On one hand polynomials are defined by the most straightforward operations. One doesn't need to involve inequalities (nor division). In this sense polynomials form a narrow class about which it is easy to prove useful theorems and algorithms. On the other hand, as @JO'R has mentioned in the Question, all continuous functions are approximated by polynomials. Thus this family is in this sense rich and useful in applications.

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Aside from any philosophical reasons, I think the pragmatic side of it is the reason they are so popular.

  • They are easy to compute and deal with.

    a) before computers: differentiation, integration are easy. Also one of reason why Fourier series are popular.

    b) after computers: splines

  • Most other functions (exponents, trig functions etc) are glorified polynomials, via Taylor series.

  • Sometimes undeserved popularity: how many times have you seen a 15-term polynomial used in a regression equation?

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the question of usefulness is essentially, whether most functions are (at least locally) analytic, or not.

If most functions are not even locally analytic, then the usefulness of polynoms is only an illusion that comes from the fact, that the formulation of many problems implies certain degrees of continuity of the solution (e.g. differential equations).

But, what about optimization, fractals or combinatorics? Is it really possible to calculate optimal solutions, the value of a fractal or the answer to a general combinatorial problem with polynomes?

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Either I completely misunderstand your answer, or I find this to be a very weird use of the word "illusion". If many problems imply certain continuity restraints and thus local analyticity, how is it that polynomials seem useful but are not? –  rghthndsd Jun 13 at 14:16
    
@rghthndsd you did not read my answer carefully enough. I didn't say "most problems imply...", but rather "the formulation of most problems" –  Manfred Weis Jun 13 at 15:44
    
Ah, I see. So then perhaps you're suggesting: "if most functions are not locally analytic, then our formulation of many problems is somehow off". Is this more accurate? –  rghthndsd Jun 13 at 20:22
    
my point is, that the problems, that humans will ever formulate are biased by what they experience and, by their limited mental capabilitiies (it is somewhat a gerrymandering in the realm of mathematics). –  Manfred Weis Jun 16 at 9:53
    
regarding your comment: yes, in my opinion polynomials are so usefull, because they are a sufficient approximation of what we need to model. –  Manfred Weis Jun 16 at 10:09

Mathematicians and all the tools they can use are subject to the laws of physics which impose certain constraints, e.g. that you can only ever perform a finite number of elementary operations. In a universe governed by classical mechanics this is not true, there you could construct a machine whose clock cycle can increase exponentially fast. Such a machine could perform an infinite amount of computations in a finite time. It is because we don't live in such a universe that polynomials are useful to us.

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If I'm reading arXiv:1203.4667 right, there's a very precise (but maybe unrelated) connection between polynomials and computational complexity. The Turing machine model of computation is apparently equivalent, from a complexity theory perspective, to Shannon's "General Purpose Analog Computer" (GPAC) model, and a function can be computed by a GPAC if and only if it's a coordinate of a solution to a differential equation $\dot{y} = p(y)$, where $p$ is a polynomial function from $\mathbb{R}^n$ to itself. –  Vectornaut Jun 15 at 19:57

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