polynomial in nLab

Context

Algebra

• Definitions and meanings
• Properties
• As differential algebras
• As domains
• Examples
• Related concepts

Example

The set of polynomials in one variable zz with coefficients in RR, also called the set of univariate polynomials, is the set R[ℕ]R[\mathbb{N}] of all formal linear combinations on elements n∈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 n∈Ra_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 k⋅z l=z k+l. z^k \cdot z^l = z^{k+l} \,.

Definition

In the case of univariate polynomials, since the function type R[z]→R[z]R[z] \to R[z] is a function R R -algebra with the commutative R R -algebra homomorphism to constant functions C:R→(R[z]→R[z])C:R \to (R[z] \to R[z]), there exists a commutative R[z]R[z]-algebra homomorphism i:R[z]→(R[z]→R[z])i:R[z] \to (R[z] \to R[z]) inductively defined by i(s(r))≔C(r)i(s(r)) \coloneqq C(r) for r∈Rr \in R and i(z)≔id R[z]i(z) \coloneqq id_{R[z]}. The substitution or composition of univariate polynomials (−)∘(−):R[z]×R[z]→R[z](-) \circ (-):R[z] \times R[z] \to R[z] is the uncurrying of ii, p∘q≔i(p)(q)p \circ q \coloneqq i(p)(q) for p∈R[z]p \in R[z] and q∈R[z]q \in R[z].

Proposition

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

Proof

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.

Definition

For every univariate polynomial ring R[z]R[z], one could inductively define a function called a derivative or derivation

∂∂z:R[z]→R[z]\frac{\partial}{\partial z}: R[z] \to R[z]

such that

• for all polynomials p∈R[z]p \in R[z] and q∈R[z]q \in R[z] and constant polynomials a∈R[z]a \in R[z] and b∈R[z]b \in R[z],

  ∂∂z(a⋅p+b⋅q)=a⋅∂∂z(p)+b⋅∂∂t(q)\frac{\partial}{\partial z}(a \cdot p + b \cdot q) = a \cdot \frac{\partial}{\partial z}(p) + b \cdot \frac{\partial}{\partial t}(q)

• for all polynomials p∈R[z]p \in R[z] and q∈R[z]q \in R[z],

  ∂∂z(p⋅q)=p⋅∂∂z(q)+∂∂z(p)⋅q\frac{\partial}{\partial z}(p \cdot q) = p \cdot \frac{\partial}{\partial z}(q) + \frac{\partial}{\partial z}(p) \cdot q

• and

  ∂∂z(z)=1\frac{\partial}{\partial z}(z) = 1

  Thus the univariate polynomial ring R[z]R[z] is a differential algebra.

Proposition

In the multivariate polynomial ring R[z 1,z 2…z n]R[z_1, z_2 \ldots z_n], there is a derivation

∂∂z i:R[z 1,z 2…z n]→R[z 1,z 2…z n]\frac{\partial}{\partial z_i}: R[z_1, z_2 \ldots z_n] \to R[z_1, z_2 \ldots z_n]

for each 1≤i≤n1 \leq i \leq n called a partial derivative.

Proposition

Since RR is power-associative, for every positive integer n∈ℤ +n \in \mathbb{Z}_+,

∂∂zz n+1=j(n)∘z n\frac{\partial}{\partial z}z^{n+1} = j(n) \circ z^{n}

where j:ℤ +→ℤ→R[z]j:\mathbb{Z}_+ \to \mathbb{Z} \to R[z] is the canonical function from the positive integers to R[x]R[x].

Proposition

If RR is an integral domain, then for all constant polynomials a∈R[z]a \in R[z],

∂∂xa=0\frac{\partial}{\partial x}a = 0

Definition

Given a polynomial p∈R[z]p \in R[z], we define the set of antiderivatives of pp to be the fiber of the derivative at pp:

antiderivatives(p)≔{q∈R[z]|∂∂z(q)= R[z]p}\mathrm{antiderivatives}(p) \coloneqq \left\{q \in R[z] \bigg| \frac{\partial}{\partial z}\left(q\right) =_{R[z]} p\right\}

Lemma

Let RR be a commutative ring. Given f,g∈R[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,r∈k[x]q, r \in k[x] such that f=q⋅g+rf = q \cdot g + r and deg(r)<deg(g)\deg(r) \lt \deg(g).

Proof

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 m−ng(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 q⋅g+r=q′⋅g+r′q \cdot g + r = q' \cdot g + r' and q≠q′q \neq q', we have deg((q−q′)g)=deg(r′−r)<deg(g)\deg((q - q')g) = \deg(r' - r) \lt \deg(g) which is impossible; then r=r′r = r' quickly follows from q=q′q = q'.

Corollary

For any commutative ring RR, if a∈Ra \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 (x−a)q(x)(x - a)q(x).

Proof

Since x−ax - a is monic, we may write p(x)=(x−a)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.