# nLab constant polynomial

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

A constant polynomial is a polynomial where not having degree $0$ 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 $0$ or is the zero polynomial.)

Equivalently, given a commutative ring $R$, polynomial rings $R[S]$ are free commutative $R$-algebras on an inhabited finite set of generators $S$; and are characterized by the universal property that it is the initial object with a ring homomorphism $h:R \to R[S]$ and a function $i:S \to R[S]$. A constant polynomial of $R[S]$ is a polynomial which is in the image of the canonical ring homomorphism $h:R \to R[S]$. 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 $F: Set \to CommRing$ is the left adjoint of the forgetful functor $U: CommRing \to Set$, so that $F$ takes a set $S$ to the polynomial ring $R[S]$, then a polynomial in (say) $n$ variables can be naturally identified with any of the following:

• An element $1 \to U F(n)$;

• A ring map $F(1) \to F(n)$;

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

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

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