nLab constant polynomial



A constant polynomial is a polynomial where not having degree 00 implies that it is the zero polynomial. (In classical mathematics this is equivalent to saying that a constant polynomial is a polynomial which either has degree 00 or is the zero polynomial.)

Equivalently, given a commutative ring RR, polynomial rings R[S]R[S] are free commutative R R -algebras on an inhabited finite set of generators SS; and are characterized by the universal property that it is the initial object with a ring homomorphism h:RR[S]h:R \to R[S] and a function i:SR[S]i:S \to R[S]. A constant polynomial of R[S]R[S] is a polynomial which is in the image of the canonical ring homomorphism h:RR[S]h:R \to R[S]. As a result, the subring of constant polynomials is in isomorphism with the ground ring.

A related point of view, falling under the rubric “structure-semantics duality” in the old terminology of Lawvere, emanates from the recognition that if F:SetCommRingF: Set \to CommRing is the left adjoint of the forgetful functor U:CommRingSetU: CommRing \to Set, so that FF takes a set SS to the polynomial ring R[S]R[S], then a polynomial in (say) nn variables can be naturally identified with any of the following:

  • An element 1UF(n)1 \to U F(n);

  • A ring map F(1)F(n)F(1) \to F(n);

  • A natural transformation between representable functors CommRing(F(n),)CommRing(F(1),)CommRing(F(n), -) \to CommRing(F(1), -);

  • A natural transformation U nUU^n \to U.

From that point of view, a polynomial in nn variables is constant if the corresponding transformation U nUU^n \to U is constant in the sense of constant morphism, i.e., factors through the terminal functor U 0U^0.


Last revised on July 25, 2023 at 12:55:25. See the history of this page for a list of all contributions to it.