Homogeneous Substitution

Summary: When $y$ and $z$ are tied by an implicit polynomial equation in which all terms have restricted combinations of total degrees, the substitution $y=xz$ (or the more general $y=x^m{z}^n$) collapses the equation so that $z$ can be solved for in terms of the new variable $x$. This is Euler’s §52–§58 technique.

Sources: chapter3

Last updated: 2026-04-23

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The trick

Given an implicit relation $F(y,z)=0$ that cannot be solved explicitly for $y$ or $z$, introduce $x$ by

$y=xz\text{(or more generally}y=x^m{z}^n\text{).}$

Substituting into $F$ often produces an equation whose powers of $z$ can be factored out, leaving a single power of $z$ equal to a rational function of $x$. That in turn gives $z$ as a root of a rational expression in $x$, and then $y=xz$ follows.

The technique presupposes that the implicit relation has enough “homogeneity” — either literal (all terms of the same total degree in $y,z$) or arithmetic structure across the set of monomials that appear.

§52 — Three-term equation with one mixed monomial

For equations of the form

$a{y}^{\alpha }+b{z}^{\beta }+c{y}^{\gamma }{z}^{\delta }=0,$

Euler substitutes $y=x^m{z}^n$ to get

$a{x}^{\alpha m}{z}^{\alpha n}+b{z}^{\beta }+c{x}^{\gamma m}{z}^{\gamma n+\delta }=0.$

He then chooses $n$ to make two of the three exponents of $z$ equal, so they can be collected and the common power of $z$ factored out. Three choices of $n$ are available (source: chapter3, §52):

• I. $\alpha n=\beta$, giving $n=\beta /\alpha$.
• II. $\beta =\gamma n+\delta$, giving $n=(\beta -\delta )/\gamma$.
• III. $\alpha n=\gamma n+\delta$, giving $n=\delta /(\alpha -\gamma )$.

Each choice expresses $z$ and $y$ as rational powers of rational functions of $x$. Any integer choice of $m$ gives the most convenient form of the formula. The classic example is the folium-of-descartes.

§53 — A posteriori construction

Given a rational parametrization $z={\left(\frac{a{x}^{\alpha }+b{x}^{\beta }+\cdots }{A+B{x}^{\mu }+\cdots }\right)}^{p/r}$ and $y=x{z}^{q/p}$, one can reverse-engineer the implicit relation $F(y,z)=0$ it parametrizes. Euler uses $y^p=x^p{z}^q$, so $x=y{z}^{-q/p}$, and substitutes back (source: chapter3, §53). The construction is the inverse of §52.

§54–§57 — Exactly two total degrees

If the monomials of $F(y,z)$ come in exactly two total degrees $m$ and $n$ (with $m>n$), let $y=xz$. The equation becomes

$(\text{polynomial in }x)\cdot z^m+(\text{polynomial in }x)\cdot z^n=0,$

or, after dividing by $z^n$,

$z^{m-n}=\frac{\text{polynomial in }x}{\text{polynomial in }x}.$

So $z$ is an $(m-n)$-th root of a rational function of $x$.

Explicit cases Euler works out:

• §54. $a{y}^2+byz+c{z}^2+dy+ez=0$. Degrees $2$ and $1$. Gives $z=-\frac{dx+e}{a{x}^2+bx+c}$ and $y=-\frac{(dx+e)x}{a{x}^2+bx+c}$ — both rational in $x$ (source: chapter3, §54).
• §55. $a{y}^3+b{y}^{2}z+cy{z}^2+d{z}^3+e{y}^2+fyz+g{z}^2=0$. Degrees $3$ and $2$. Gives $z=-\frac{e{x}^2+fx+g}{a{x}^3+b{x}^2+cx+d}$, and $y=xz$ (source: chapter3, §55).
• §56. $a{y}^2+byz+c{z}^2=d$. Degrees $2$ and $0$. Gives $z=\sqrt{d/(a{x}^2+bx+c)}$, $y=x\sqrt{d/(a{x}^2+bx+c)}$ (source: chapter3, §56).
• §57. General case, degrees $m$ and $n$: $z={\left(\frac{\alpha x^n+\beta x^{n-1}+\cdots }{a{x}^m+b{x}^{m-1}+\cdots }\right)}^{1/(m-n)}$ (source: chapter3, §57).

§58 — Three total degrees in arithmetic progression

If the monomials of $F$ have exactly three total degrees $m_1<m_2<m_3$ with $m_3-m_2=m_2-m_1$, let $y=xz$ and divide by $z^{m_1}$. The equation becomes a quadratic in $z^{m_2-m_1}$, solvable by the quadratic formula (source: chapter3, §58).

Example: $a{y}^3+b{y}^{2}z+cy{z}^2+d{z}^3=2e{y}^2+2fyz+2g{z}^2+hy+jz$ has total degrees $3,2,1$. After $y=xz$ and dividing by $z$,

$(a{x}^3+b{x}^2+cx+d)z^2=2(e{x}^2+fx+g)z+(hx+j),$

and the quadratic formula gives $z$.

Why this matters

These are the first systematic algebraic parametrizations in Euler’s treatise. They anticipate the modern observation that a plane algebraic curve of genus zero admits a rational parametrization, and that a genus-zero curve with two “singular” behaviours at infinity (e.g. a nodal cubic like the folium-of-descartes) can be parametrized by projecting from the singular point — which is exactly what $y=xz$ does when the equation is homogeneous enough.

Theoretical justification (Chapter 5, §88)

In Chapter 3 Euler uses $y=xz$ as a procedure that “happens to work” when the equation is homogeneous enough. The theorem that justifies it appears only in Chapter 5, §88 (see homogeneous-function):

If $V(y,z)$ is homogeneous of degree $n$, then under $y=uz$ we have $V=z^n\cdot f(u)$, where $u=y/z$.

This is why the §54–§57 cases all reduce cleanly to ”$z^{m-n}=$ rational function of $x$”: the left-hand side of $F(y,z)=0$ splits into homogeneous pieces of degrees $m$ and $n$, each piece becomes $z^{\mathrm{deg}}\cdot (\text{poly in }x)$ under $y=xz$, and equating gives an equation in $z$ whose powers are $m$ and $n$ alone — so $z^{m-n}$ is determined by a rational function of $x$.

The degree-zero case §89 (where $V$ becomes a function of $u$ alone, with the $z$-factor disappearing) corresponds to §56’s totally homogeneous equation $a{y}^2+byz+c{z}^2=d$, where the ratio $y/z$ is not enough to pin down $z$ and an extra radical survives.

Related pages

• substitution
• chapter-3-on-the-transformation-of-functions-by-substitution
• folium-of-descartes
• rationalizing-substitutions
• rational-parametrization-of-the-circle
• homogeneous-function
• chapter-5-on-functions-of-two-or-more-variables