Monomial on $\mathbb{R}^n$ ¶

Definition¶

A monomial on $\mathbb{R}^n$ is a function from $\mathbb{R}^n$ to $\mathbb{R}$ given by a product of real coefficients and variables raised to nonnegative integer powers, i.e., a monomial has the general expression

f(x_1,x_2,\dots,x_n) = a x_1^{k_1} x_2^{k_2} \cdots x_n^{k_n},

with constant $a \in \mathbb{R}$ and nonnegative integer exponents $k_1,\, \dots,\, k_n$ . When $a\ne0$ $0$ , its total degree is the sum $k_1+\cdots+k_n$ $k_{n}$ . When $a=0$ $0$ , it is the zero function, whose total degree is left undefined; the same zero function can be written with any exponent tuple.

For $n=1$ this is the usual Monomial; for several variables it is a Function between Spaces from an input tuple to one real output.

Interpretation¶

A monomial on $\mathbb{R}^n$ is the simplest building block of a polynomial in several variables: a constant times a product of powers of the variables, such as $-2x^2y$ . Its total degree counts the variable factors in that product with repetition, so $x^2y=x\cdot x\cdot y$ has total degree $3$ .

The exponent tuple $(k_1,\dots,k_n)$ records how often each variable occurs, and an exponent $0$ means that the variable is absent: on $\mathbb{R}^3$ , $x_1^4$ is the monomial with exponents $(4,0,0)$ . The total degree differs from the degree in a single variable: $x^2y$ has degree $2$ in $x$ , degree $1$ in $y$ , and total degree $3$ . Scaling all variables by $c$ shows why the total degree matters:

a(cx_1)^{k_1}\cdots(cx_n)^{k_n}=c^{k_1+\cdots+k_n}\,ax_1^{k_1}\cdots x_n^{k_n},

so the total degree is the degree of homogeneity (Key Results).

A common slip is to allow other exponents. The exponents must be nonnegative integers, so $x^{1/2}y^{1/2}$ and $x^{-1}y$ are not monomials, although on $(0,\infty)^2$ they are homogeneous functions of degrees $1$ and $0$ .

For $n=1$ a monomial is the familiar monomial $ax^k$ . Finite sums of monomials are the polynomials on $\mathbb{R}^n$ , and sums of monomials of one total degree are homogeneous polynomials. The monomials of total degree $2$ are the multiples $ax_ix_j$ of the products of two variables, the terms of a quadratic form.

Key Results¶

The product of two monomials is again a monomial. The exponents add componentwise, and the constant multiplies.
Every nonzero monomial is a homogeneous function whose degree equals its total degree. The zero function satisfies the homogeneity identity for every degree.

Examples¶

$f(x,y)=-2x^2y$ is a monomial with exponents $k_1=2$ $2$ , $k_2=1$ $1$ , total degree $3$ .
The monomial $f(x,y)=x^4$ has $k_1=4$ $4$ , $k_2=0$ $0$ , total degree $4$ .

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 7

Aliases: monomial in more than one indeterminate, monomial in more than one variable, multivariate monomial Symbols: element of, infinity, real numbers

Last updated: 27 October 2026 11:52

Monomial on Rn\mathbb{R}^nRn¶