Definitions and meanings

Let RR be a commutative ring. A polynomial with coefficients in RR is an element of a polynomial ring over RR. A polynomial ring over RR consists of a set XX whose elements are called “variables” or “indeterminates”, and a function XR[X]X \to R[X] to (the underlying set of) a commutative RR-algebra that is universal among such functions, so that R[X]R[X] is the free commutative RR-algebra generated by XX; a polynomial is then an element of the underlying set of R[X]R[X].


Much like “vector”, the term “polynomial” in this sense may seem slightly deprecated from the viewpoint of modern mathematics. We no longer think of a vector space as consisting of things called “vectors” (i.e. we don’t assign an objective meaning to “vectors”); it’s the other way around, where we introduce a type of structure called a vector space and then, relative to a given vector space context, declare that “vector” just means an element therein. Similarly, the term “polynomial” in the sense above has seemingly been subordinated to the structural concept of polynomial ring. (In Linderholm’s Mathematics Made Difficult, page 152, there is an amusing passage where someone points at the expression a 0+a 1X+a 2X 2++a nX na_0 + a_1 X + a_2 X^2 + \ldots + a_n X^n and asks “Well, how about it? Is it a polynomial, or isn’t it?” and the respondent says, “Yeah, sure, I guess. Looks like one. Yeah, sure, that’s a polynomial all right” – an answer which is not wrong exactly, but not quite right either, since it fails to recognize that there is something questionable about the question.)


On further reflection, however, we might more objectively identify the concept of “polynomial” (let us say a polynomial in nn variables) with a definable nn-ary operation in the theory of commutative RR-algebras. From a categorical perspective, if U:CommAlg RSetU: CommAlg_R \to Set is the forgetful functor, a definable nn-ary operation means a natural transformation U nUU^n \to U. The connection is that the functor U nU^n is representable, being represented by the free object F[n]=R[x 1,,x n]F[n] = R[x_1, \ldots, x_n], so that a natural transformation U nUU^n \to U is canonically identified with a transformation hom(R[x 1,,x n],)U\hom(R[x_1, \ldots, x_n], -) \to U or to an element of UR[x 1,,x n]U R[x_1, \ldots, x_n], by the Yoneda lemma. In pursuit of this objective meaning (which is essentially due to Lawvere), we find that the “variable” x ix_i stands for the i thi^{th} projection map U nUU^n \to U, and that the meaning of (let’s say) x 1x 2R[x 1,x 2,x 3]x_1 x_2 \in R[x_1, x_2, x_3] is that it is the definable operation whose instantiation at any commutative RR-algebra AA is the function A 3AA^3 \to A taking (a,b,c)(a, b, c) to aba b.

Of course there are traditional standard expressions that people usually have in mind when they speak of “a polynomial” as such. But leaving it at that, where polynomials are merely identified with certain types of expressions (as by the characters in Linderholm’s book), ignores the deeper objective meaning of definable operations which of course is the actual point of it all.


Finally, sometimes “polynomial” is construed to mean a polynomial function. This is actually just a particular instantiation of a definable operation. The default meaning is that, if we are working for instance with a polynomial ring in one variable R[x]R[x], then we have a composite

UR[x]hom(U,U)ev Rhom(UR,UR)U R[x] \cong \hom(U, U) \stackrel{ev_R}{\to} \hom(U R, U R)

where the second map sends a natural transformation to its component at RR as an RR-algebra. Put differently, the set hom(UR,UR)\hom(U R, U R) carries a commutative RR-algebra structure under the pointwise operations, and there is a unique RR-algebra map R[x]hom(UR,UR)R[x] \to \hom(U R, U R) that sends ‘xx’ to the identity map. The value? of a polynomial pR[x]p \in R[x] under this map is then the corresponding polynomial function URURU R \to U R.

This conflation of polynomials with polynomial functions is often forgivable, particularly in those cases where the map R[x]hom(UR,UR)R[x] \to \hom(U R, U R) is injective (so that ‘polynomials’ are identified with certain types of functions). Of course the map won’t be injective if RR is finite, to give one example. But in analysis, where we consider functions on \mathbb{R} or \mathbb{C}, the conflation is familiar and rarely cause for concern. The conflation may also be responsible for certain notational artifacts, such as the common (and useful!) practice of writing p(x)p(x) for polynomials pp.

With these preliminary remarks out of the way, we recall some of the more syntactic considerations with an example.


The set of polynomials in one variable zz with coefficients in RR is the set R[]R[\mathbb{N}] of all formal linear combinations on elements nn \in \mathbb{N}, thought of as powers z nz^n of the variable zz. As a string of symbols, a polynomial is frequently represented by a form like

a nz n++a 1z+a 0, a_n z^n + \cdots + a_1 z + a_0 \,,

where nn is an arbitrary natural number and a 0,,a nRa_0, \dots, a_n \in R, subject to the usual fine print (where we work modulo the congruence generated by equations of the form

0z n+1+a nz n++a 1z+a 0=a nz n++a 1z+a 0 0 z^{n+1} + a_n z^n + \cdots + a_1 z + a_0 = a_n z^n + \cdots + a_1 z + a_0

so that we ignore coefficients of zero). (The degree of a polynomial is the maximum nn for which a na_n is nonzero, in which case the leading term of the polynomial is a nz na_n z^n. A polynomial is constant if its degree is 00. The degree of the zero polynomial may be left chastely undefined, although for some purposes it may be convenient to define it as 1-1 or as -\infty. Even 00 is possible if one is prepared to observe some fine print. Chacun à son goût.)

This set is equipped with an RR-module structure (where formal linear combinations are added and scalar-multiplied as usual) and also with the structure of a ring, in fact a commutative algebra over RR, denoted R[z]R[z] and called the polynomial ring or ring of polynomials, with ring multiplication

R[z]R[z]R[z] R[z] \cdot R[z] \to R[z]

the unique one that bilinearly extends the multiplication of monomials given by

z kz l=z k+l. z^k \cdot z^l = z^{k+l} \,.

Thus, one way to construct a polynomial ring is first to construct the free commutative monoid generated by a set XX (the monoid of monomials), and then to construct the free RR-module generated by the underlying set of that monoid, extending the monoid multiplication to a ring multiplication by bilinearity.

In addition to the ring structure, there is a further operation R[z]×R[z]R[z]R[z] \times R[z] \to R[z] which may be described as “substitution”; see Remark 5 below for a general description (which applies in fact to any Lawvere theory).

Moreover, there is a noncommutative analogue of polynomial ring on a set XX, efficiently described as the free RR-module generated by the (underlying set of the) free monoid on XX. This carries also a ring structure, with ring multiplication induced from the monoid multiplication. A far-reaching generalization of this construction is given at distributive law.

Finally: polynomial algebras may be regarded as graded algebras (graded over \mathbb{N}). Specifically: let us regard R[X]R[X] as the free RR-module generated by (the underlying set of) the free commutative monoid F(X)F(X). The monoid homomorphism F(!):F(X)F(1)F(!): F(X) \to F(1) \cong \mathbb{N} induced by the unique function !:X1!: X \to 1 gives an \mathbb{N}-fibering of F(X)F(X) over \mathbb{N}, with typical fiber F(X) nF(X)_n whose elements are called monomials of degree nn. Then the homogeneous component of degree nn in R[X]R[X] is the RR-submodule generated by the subset F(X) nF(X)F(X)_n \subset F(X). The elements of this component are called homogeneous polynomials of degree nn.



The polynomial ring R[z]R[z] is the free RR-algebra on one generator (the variable zz).


By the definition of free objects one needs to check that algebra homomorphisms

f:R[z]K f : R[z] \to K

to another algebra K are in natural bijection with functions of sets

f¯:*K \bar f : * \to K

from the singleton to the set underlying KK. Take f¯f(z)\bar f \coloneqq f(z). Using RR-linearity, this is directly seen to yield the desired bijection.


Similarly, the set of polynomials in any given set of variables with coefficients in RR is the free commutative RR-algebra on that set of generators; see symmetric power and symmetric algebra.


As usual in the study of universal algebra via Lawvere theories, there is an operad whose n thn^{th} component C nC_n is the free algebra R[x 1,,x n]R[x_1, \ldots, x_n], and whose operadic multiplication is given by maps

C k×C n 1××C n kC nC_k \times C_{n_1} \times \ldots \times C_{n_k} \to C_n

(n=n 1++n kn = n_1 + \ldots + n_k) that take a tuple of elements (p;q 1,,q k)(p; q_1, \ldots, q_k) to p(q 1(x),,q k(x))p(q_1(x), \ldots, q_k(x)). Formally, it takes this tuple to the value of pp under the unique algebra map R[x 1,,x k]R[x 1,,x n]R[x_1, \ldots, x_k] \to R[x_1, \ldots, x_n] that extends the mapping x ji j(q j)x_j \mapsto i_j(q_j). Here the i j:R[x 1,,x n j]R[x 1,,x n]i_j: R[x_1, \ldots, x_{n_j}] \to R[x_1, \ldots, x_n] are appropriate coproduct inclusions (in the category of commutative rings), where i j(x l)=x n 1++n j1+li_j(x_l) = x_{n_1 + \ldots + n_{j-1} + l}. A particularly important case of substitution is the case k=1k=1 and n 1=1n_1 = 1, where the map R[x]×R[x]R[x]R[x] \times R[x] \to R[x] is ordinary substitution (p,q)p(q(x))(p, q) \mapsto p(q(x)). This is a special case of the more general notion of Tall-Wraith monoid.


In case RR is an integral domain, the field of fractions of R[z]R[z] is the field R(z)R(z) of rational functions.

Polynomials in one variable

In the case where R=kR = k is a field, the polynomial ring k[x]k[x] has a number of useful properties. One is that it is a Euclidean domain, where the degree serves as the Euclidean function:


Let RR be a commutative ring. Given f,gR[x]f, g \in R[x] where the leading coefficient of gg is a unit (e.g., if gg is a monic polynomial), there are unique q,rk[x]q, r \in k[x] such that f=qg+rf = q \cdot g + r and deg(r)<deg(g)\deg(r) \lt \deg(g).


If deg(f)<deg(g)\deg(f) \lt \deg(g), then q=0q = 0 and r=fr = f will serve. Otherwise we may argue by induction on deg(f)\deg(f), where if a mx ma_m x^m is the leading term of ff and b nx nb_n x^n the leading term of gg, then h(x)=f(x)a mb n 1x mng(x)h(x) = f(x) - a_m b_n^{-1} x^{m-n}g(x) has lower degree than f(x)f(x). This proves existence. Uniqueness is clear, since if qg+r=qg+rq \cdot g + r = q' \cdot g + r' and qqq \neq q', we have deg((qq)g)=deg(rr)<deg(g)\deg((q - q')g) = \deg(r' - r) \lt \deg(g) which is impossible; then r=rr = r' quickly follows from q=qq = q'.


If kk is field, then k[x]k[x] is a Euclidean domain. As a result, k[x]k[x] is a principal ideal domain, and therefore a unique factorization domain.

See Euclidean domain for a proof.


For any commutative ring RR, if aRa \in R is a root of p(x)R[x]p(x) \in R[x], i.e., if the value of the polynomial function p(a)p(a) is 00, then p(x)p(x) is of the form (xa)q(x)(x - a)q(x).


Since xax - a is monic, we may write p(x)=(xa)q(x)+rp(x) = (x - a)q(x) + r where deg(r)<1\deg(r) \lt 1, whence rr is a constant. By evaluating the polynomial function at x=ax = a, the term rr is 00.

This observation may be exploited in various neat ways. One is that if p(x)p(x) is a polynomial, then p(y)=p(x)+(yx)q(x,y)p(y) = p(x) + (y - x)q(x, y) for some unique q(x,y)k[x,y]q(x, y) \in k[x, y]. A consequence is that the Lawvere theory of commutative kk-algebras is a Fermat theory. The derivative of pp may be defined to be q(x,x)k[x]q(x, x) \in k[x].

Revised on July 4, 2015 10:11:06 by Todd Trimble (