nLab real polynomial function

Definition

Without scalar coefficients

A real polynomial function is a polynomial function in the real numbers, a function $f:\mathbb{R} \to \mathbb{R}$ such that

• $f$ is in the image of the function $j:\mathbb{R}^* \to (\mathbb{R} \to \mathbb{R})$ from the free monoid $\mathbb{R}^*$ on $\mathbb{R}$, i.e. the set of lists of real numbers, to the function algebra $\mathbb{R} \to \mathbb{R}$, such that

  • $j(\epsilon) = 0$, where $0$ is the zero function.
  • for all $a \in \mathbb{R}^*$ and $b \in \mathbb{R}^*$, $j(a b) = j(a) + j(b) \cdot (-)^{\mathrm{len}(a)}$, where $(-)^n$ is the $n$-th power function for $n \in \mathbb{N}$
  • for all $r \in \mathbb{R}$, $j(r) = c_r$, where $c_r$ is the constant function whose value is always $r$.

• $f$ is in the image of the canonical ring homomorphism $i:\mathbb{R}[x] \to (\mathbb{R} \to \mathbb{R})$ from the real polynomial ring in one indeterminant $\mathbb{R}[x]$ to the function algebra $\mathbb{R} \to \mathbb{R}$, which takes constant polynomials in $\mathbb{R}[x]$ to constant functions in $\mathbb{R} \to \mathbb{R}$ and the indeterminant $x$ in $\mathbb{R}[x]$ to the identity function $\mathrm{id}_\mathbb{R}$ in $\mathbb{R} \to \mathbb{R}$

• There exists a natural number $n$ such that the $n$-th order derivative of $f$ is equal to the zero function:

  $$\frac{d^n f}{d x^n} = 0$$

With scalar coefficients

A real polynomial function is a function $f:\mathbb{R} \to \mathbb{R}$ with a natural number $n \in \mathbb{N}$ and a list of length $n$ of real numbers $a:[0, n)_\mathbb{N} \to \mathbb{R}$ which satisfy one of these conditions:

• for all $x \in R$,

  $$f(x) = \sum_{i:[0, n)_\mathbb{N}} a(i) \cdot x^i$$

  where $x^i$ is the $i$-th power function for multiplication.

• $f$ is a solution to the $n$-th order linear homogeneous ordinary differential equation

  $$\frac{d^n f}{d x^n} = 0$$

  with initial conditions

  $$\frac{d^i f}{d x^i}(0) = i! \cdot a(i)$$

  for each natural number $i:[0, n)_\mathbb{N}$

Properties

Degree

Without scalar coefficients

Given a non-zero real polynomial function $f:\mathbb{R} \to \mathbb{R}$, the degree of $f$ is the maximum natural number $n$ where the $n$-th derivative of $f$ is not the zero-function:

$$\mathrm{deg}(f) \coloneqq \max_{n \in \mathbb{N}, \frac{d^{n} f}{d x^{n}} \neq 0}(n)$$

With scalar coefficients

The degree of a real polynomial function, with additional data of a natural number $n \in \mathbb{N}$ and a list of length $n$ of real numbers $a:[0, n)_\mathbb{N} \to \mathbb{R}$, is defined as the maximum natural number $i \lt n$ such that $\vert a_i \vert \gt 0$.

In constructive mathematics

In constructive mathematics, there exist real polynomial functions for which one cannot prove that a particular natural number $i \lt n$ is the degree of the real polynomial function: i.e. the function sub-$\mathbb{R}$-algebra of real polynomial functions is not a Euclidean domain.

Pointwise continuity

Given a real polynomial function $f:\mathbb{R} \to \mathbb{R}$, $f$ is a pointwise continuous function with respect to its metric topology defined through the absolute value function, subtraction, and its strict total order.

Pointwise differentiability

Given a real polynomial function $f:\mathbb{R} \to \mathbb{R}$, $f$ is a pointwise differentiable function with respect to its metric topology defined through the absolute value function, subtraction, and its strict total order.

Smoothness

Every real polynomial function is a smooth function. This could be shown coinductively: A smooth function is a pointwise differentiable function whose derivative is also smooth, and thus the $n$-th derivative of a smooth function is a smooth function. The zero function $f(x) = 0$ is a smooth function, every real polynomial function is a pointwise differentiable function and the $n$-th derivative of a polynomial function, with a natural number $n \in \mathbb{R}$ and a list of length $n$ of real numbers $a:[0, n)_\mathbb{N} \to \mathbb{R}$ such that one of the two conditions above is satisfied, is the zero function. Thus, every polynomial function is a smooth function.

See also

• real number

• polynomial function

• real quadratic function

• real cubic function

• rational functions are continuous

References

The proof of pointwise continuity using epsilontic analysis is spelled out for instance in

• Kyle Miller, Polynomials are continuous functions, 2014 (pdf)

See also:

• Wikipedia, Polynomial function