symmetric monoidal (∞,1)-category of spectra
Let be a commutative ring. A polynomial with coefficients in is an element of a polynomial ring over . A polynomial ring over consists of a set whose elements are called “variables” or “indeterminates”, and a function to (the underlying set of) a commutative -algebra that is universal among such functions, so that is the free commutative -algebra generated by ; a polynomial is then an element of the underlying set of .
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 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 variables) with a definable -ary operation in the theory of commutative -algebras. From a categorical perspective, if is the forgetful functor, a definable -ary operation means a natural transformation . The connection is that the functor is representable, being represented by the free object , so that a natural transformation is canonically identified with a transformation or to an element of , by the Yoneda lemma. In pursuit of this objective meaning (which is essentially due to Lawvere), we find that the “variable” stands for the projection map , and that the meaning of (let’s say) is that it is the definable operation whose instantiation at any commutative -algebra is the function taking to .
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 , then we have a composite
where the second map sends a natural transformation to its component at as an -algebra. Put differently, the set carries a commutative -algebra structure under the pointwise operations, and there is a unique -algebra map that sends ‘’ to the identity map. The value? of a polynomial under this map is then the corresponding polynomial function .
This conflation of polynomials with polynomial functions is often forgivable, particularly in those cases where the map is injective (so that ‘polynomials’ are identified with certain types of functions). Of course the map won’t be injective if is finite, to give one example. But in analysis, where we consider functions on or , 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 for polynomials .
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 with coefficients in is the set of all formal linear combinations on elements , thought of as powers of the variable . As a string of symbols, a polynomial is frequently represented by a form like
so that we ignore coefficients of zero). (The degree of a polynomial is the maximum for which is nonzero, in which case the leading term of the polynomial is . A polynomial is constant if its degree is . The degree of the zero polynomial may be left chastely undefined, although for some purposes it may be convenient to define it as or as . Even is possible if one is prepared to observe some fine print. Chacun à son goût.)
This set is equipped with an -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 , denoted and called the polynomial ring or ring of polynomials, with ring multiplication
the unique one that bilinearly extends the multiplication of monomials given by
Thus, one way to construct a polynomial ring is first to construct the free commutative monoid generated by a set (the monoid of monomials), and then to construct the free -module generated by the underlying set of that monoid, extending the monoid multiplication to a ring multiplication by bilinearity.
Moreover, there is a noncommutative analogue of polynomial ring on a set , efficiently described as the free -module generated by the (underlying set of the) free monoid on . 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 ). Specifically: let us regard as the free -module generated by (the underlying set of) the free commutative monoid . The monoid homomorphism induced by the unique function gives an -fibering of over , with typical fiber whose elements are called monomials of degree . Then the homogeneous component of degree in is the -submodule generated by the subset . The elements of this component are called homogeneous polynomials of degree .
from the singleton to the set underlying . Take . Using -linearity, this is directly seen to yield the desired bijection.
() that take a tuple of elements to . Formally, it takes this tuple to the value of under the unique algebra map that extends the mapping . Here the are appropriate coproduct inclusions (in the category of commutative rings), where . A particularly important case of substitution is the case and , where the map is ordinary substitution . This is a special case of the more general notion of Tall-Wraith monoid.
Let be a commutative ring. Given where the leading coefficient of is a unit (e.g., if is a monic polynomial), there are unique such that and .
If , then and will serve. Otherwise we may argue by induction on , where if is the leading term of and the leading term of , then has lower degree than . This proves existence. Uniqueness is clear, since if and , we have which is impossible; then quickly follows from .
See Euclidean domain for a proof.
For any commutative ring , if is a root of , i.e., if the value of the polynomial function is , then is of the form .
Since is monic, we may write where , whence is a constant. By evaluating the polynomial function at , the term is .
This observation may be exploited in various neat ways. One is that if is a polynomial, then for some unique . A consequence is that the Lawvere theory of commutative -algebras is a Fermat theory. The derivative of may be defined to be .