Homogeneous Function

Summary: A function of $y,z$ is homogeneous of degree $n$ if every term has the same total degree $n$. This extends cleanly to rational ($\mathrm{deg}P-\mathrm{deg}Q$), irrational ($P^{\mu /\nu }$ has degree $(\mu /\nu )n$), and implicit algebraic cases. Euler’s central theorem (§88): under $y=uz$, any bivariate homogeneous function of degree $n$ becomes $z^n\cdot f(u)$; degree zero means $V$ is a function of $u=y/z$ alone. Every bivariate homogeneous polynomial factors into linear pieces $\alpha y+\beta z$, a property that fails in three or more variables.

Sources: chapter5

Last updated: 2026-04-23

---

Degree

A variable has degree 1; a constant has degree 0; the degree of a product is the sum of degrees. So $\alpha y,\beta z$ are degree 1; $\alpha y^2,\beta yz,\gamma z^2$ are degree 2; and so on (source: chapter5, §83).

A homogeneous function is one in which every term has the same degree. A heterogeneous function is one with at least two different degrees among its terms — see heterogeneous-function.

Polynomial case (§84)

Homogeneous polynomials in $y,z$ of each degree have the obvious general form:

$$\begin{array}{rl}\text{degree }1:& \alpha y+\beta z\\ \text{degree }2:& \alpha y^2+\beta yz+\gamma z^2\\ \text{degree }3:& \alpha y^3+\beta y^{2}z+\gamma y{z}^2+\delta z^3\\ \text{degree }4:& \alpha y^4+\beta y^{3}z+\gamma y^2{z}^2+\delta y{z}^3+ϵz^4\end{array}$$

Constant functions count as degree zero (source: chapter5, §84).

Rational case (§85)

A rational function $P/Q$ is homogeneous iff both $P$ and $Q$ are homogeneous, and its degree is $\mathrm{deg}P-\mathrm{deg}Q$. Negative and zero degrees are admitted (source: chapter5, §85):

function	degree
$(a{y}^2+b{z}^2)/(\alpha y+\beta z)$	1
$(y^5+z^5)/(y^2+z^2)$	3
$(y^3+z^3)/(y^{2}z)$	0
$y/z^2$	$-1$
$(y+z)/(y^4+z^4)$	$-3$
$1/(y^5+ay{z}^4)$	$-5$

Sums of homogeneous pieces of matching degree are again homogeneous of that degree: $\alpha y+\beta z^2/y+(\gamma y^4-\delta z^4)/(y^{2}z+y{z}^2)$ has degree 1 throughout.

Irrational case (§86)

If $P$ is homogeneous of degree $n$, then $P^{\mu /\nu }$ is homogeneous of degree $(\mu /\nu )n$ (source: chapter5, §86). Examples:

• $\sqrt{y^2+z^2}$ has degree 1.
• $(y^9+z^9{)}^{1/3}$ has degree 3.
• $(yz+z^2{)}^{1/2}$ has degree 1.
• $\sqrt{y^2+z^2}/\sqrt{y^4+z^4}$ has degree 0.

Euler’s sample worked expression

$\frac{1}{y}+\frac{y\sqrt{y^2+z^2}}{z^3}-\frac{y}{(y^6-z^6{)}^{1/3}}+\frac{y\sqrt{z}}{z^2\sqrt{y}+\sqrt{y^6+z^6}}$

is homogeneous of degree $-1$: each of the four summands evaluates to degree $-1$ under the rules.

Implicit case (§87)

If $V$ satisfies the polynomial equation

$V^k+P{V}^{k-1}+Q{V}^{k-2}+\cdots +R=0$

with $P,Q,\dots ,R$ polynomials in $y,z$, then $V$ is homogeneous of degree $n$ iff

$\mathrm{deg}P=n,\mathrm{deg}Q=2n,\dots ,\mathrm{deg}R=kn$

— i.e. each coefficient carries exactly the degree needed so every term of the equation is of total degree $kn$ (source: chapter5, §87). Example: $V^5+(y^4+z^4)V^3+a{y}^{8}V-z^{10}=0$ has $V$ homogeneous of degree 2.

§88 — Euler’s reduction theorem

Theorem. If $V(y,z)$ is homogeneous of degree $n$, then under $y=uz$,

$\boxed{V(y,z)=z^n\cdot f(u),u=y/z.}$

Argument. Every term of $V$ has joint degree $n$ in $y,z$. Replacing $y$ by $uz$ converts joint degree into degree in $z$ alone, so each term acquires the factor $z^n$, leaving a function of $u$ (source: chapter5, §88).

Euler checks the three cases:

• Polynomial. $V=\alpha y^3+\beta y^{2}z+\gamma y{z}^2+\delta z^3\Rightarrow V=z^3(\alpha u^3+\beta u^2+\gamma u+\delta )$.
• Rational. $V=\frac{\alpha y+\beta z}{y^2+z^2}$ (degree $-1$) $\Rightarrow V=z^{-1}\frac{\alpha u+\beta }{u^2+1}$.
• Irrational. $V=\frac{y+\sqrt{y^2+z^2}}{z\sqrt{y^3+z^3}}$ (degree $-3/2$) $\Rightarrow V=z^{-3/2}\frac{u+\sqrt{u^2+1}}{\sqrt{u^3+1}}$.

This theorem is the theoretical foundation for the parametrization trick Euler used empirically in Chapter 3 — see homogeneous-substitution.

§89 — Degree zero

If $n=0$, the factor $z^n=1$ vanishes from the expression, so

$V\text{ homogeneous of degree }0⟹V\text{ is a function of }u=y/z\text{ alone}$

(source: chapter5, §89). Examples:

• $(y+z)/(y-z)=(u+1)/(u-1)$.
• $(y-\sqrt{y^2-z^2})/z=u-\sqrt{u^2-1}$.

§90–§91 — Linear factorization in two variables

Theorem. A homogeneous polynomial of degree $n$ in $y,z$ factors as a product of $n$ linear pieces $\alpha y+\beta z$ (with real or complex coefficients) (source: chapter5, §90–§91).

Proof. By §88, the polynomial becomes $z^n\cdot p(u)$ after $y=uz$. By the fundamental-theorem-of-algebra, $p(u)$ factors into linear pieces $\alpha u+\beta$; multiplying each by $z$ recovers $\alpha y+\beta z$, giving $n$ factors in the original variables.

Corollaries:

• $a{y}^2+byz+c{z}^2$ has two linear factors.
• $a{y}^3+b{y}^{2}z+cy{z}^2+d{z}^3$ has three.
• Every homogeneous bivariate polynomial is reducible.

This fails in three or more variables. The general degree-2 homogeneous form $a{y}^2+byz+c{z}^2+dyx+ezx+f{x}^2$ does not generally split as $(\alpha y+\beta z+\gamma x)(\delta y+ϵz+\zeta x)$, and the situation is worse for higher degree (source: chapter5, §91).

Why this matters

The $y=uz$ reduction is Euler’s first structure theorem for multivariate functions. It separates scale (the $z^n$ factor) from shape (the profile $f(u)$), and it reduces the study of homogeneous bivariate functions to single-variable analysis. Geometrically it corresponds to projecting the locus $V=0$ from the origin — every line through the origin intersects the curve in a finite set, parametrized by the ratio $u=y/z$. This is the algebraic shadow of projective geometry, and it is the reason the substitution trick of Chapter 3 works for curves like the folium-of-descartes.

In modern language, §88 says a homogeneous function on ${\mathbb{R}}^2\setminus \{0\}$ of degree $n$ is determined by its restriction to the line $\{z=1\}$ (i.e. by the profile $f(u)=V(u,1)$) together with its degree. The degree-zero case §89 is the same as saying degree-zero functions descend to functions on the projective line ${\mathbb{P}}^1$.

Related pages

• functions-of-several-variables
• heterogeneous-function
• homogeneous-substitution
• reducible-polynomial
• fundamental-theorem-of-algebra
• folium-of-descartes
• chapter-5-on-functions-of-two-or-more-variables