Branches at Infinity

Summary: A curve $P+Q+R+S+\cdots =0$ (members listed by descending degree, $P$ of degree $n$) has a branch going to infinity iff the highest member $P$ contains at least one real linear factor of the form $ay-bx$. Complex linear factors pair into real quadratic factors $A^2{y}^2-2ABxy\cos \varphi +B^2{x}^2$ which never vanish even at infinity, hence cannot license an infinite branch.

Sources: chapter7 (§§166-169)

Last updated: 2026-04-28

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What “branch going to infinity” means

§166. A curve has a branch going to infinity when, for points on the curve arbitrarily far from the origin, at least one of $x$ or $y$ becomes infinite — otherwise the distance $\sqrt{x^2+y^2}$ would stay finite, contradicting “at infinity.” Either an infinite abscissa is paired with a real (finite or infinite) ordinate, or an infinite ordinate is paired with some real abscissa. So the analytic question “does the curve escape to infinity?” reduces to “for what infinite values of $x$ or $y$ does the equation continue to hold?”

Members of the equation

§170 fixes the vocabulary used throughout the chapter. Order the equation by descending degree:

$\underset{\mathrm{deg}n}{\underbrace{P}}+\underset{\mathrm{deg}n-1}{\underbrace{Q}}+\underset{\mathrm{deg}n-2}{\underbrace{R}}+\underset{\mathrm{deg}n-3}{\underbrace{S}}+\cdots =0,$

so $P$ is the highest member, $Q$ the second member, etc., down to the constant term. $P$ is a homogeneous form in $x,y$ of degree $n$, and so is uniquely determined by the equation’s leading-degree terms.

Real linear vs. complex quadratic factors

§167. Any homogeneous form of degree $n$ in two variables splits into linear factors over $\mathbb{C}$, of the form $Ay+Bx$. Real factors stay real; complex factors come in conjugate pairs $(Ay+Bx)$ and $(\bar{A}y+\bar{B}x)$ that multiply to give a real quadratic factor of the form

$A^2{y}^2-2ABxy\cos \varphi +B^2{x}^2\text{with }A,B\ne 0.$

The trigonometric coefficient comes from the modulus-argument expansion of conjugate pairs. The key inequality is

$2|AB||xy|<A^2{y}^2+B^2{x}^2\text{for all real }x,y\text{ not both zero}$

(AM-GM), so $|2ABxy\cos \varphi |<A^2{y}^2+B^2{x}^2$. Hence the real quadratic factor is strictly positive wherever it is evaluated except at $x=y=0$, and at infinity it grows as ${\infty }^2$. It cannot vanish at infinity, nor take any finite value there — it always blows up to ${\infty }^2$.

The criterion

§168. Suppose $P$ has no real linear factor — necessarily $n$ is even, and $P$ is a product of real quadratic factors of the kind above. Then as $x$ or $y$ goes to infinity, $P$ grows like ${\infty }^n$, so the equation $P+Q+R+\cdots =0$ cannot hold: lower members are of degree $<n$ and cannot cancel a strictly ${\infty }^n$ leading term.

§169. The contrapositive is the working criterion. If the equation $P+Q+R+S+\cdots =0$ has any real solutions at infinity — any branch going to infinity at all — then $P$ must contain a real linear factor $ay-bx$. The whole curve is otherwise bounded (contained in some ellipse-like region).

Specialised to a second-order line, this recovers the conic classification (classification-of-conics): the highest member is $\alpha y^2+\beta xy+\gamma x^2$, which has no real linear factor exactly when ${\beta }^2<4\alpha \gamma$ — and that is exactly the ellipse case.

Why this is the right invariant

The chapter’s central observation is that the infinite-end behaviour of the curve is decided entirely by $P$: as $|x|,|y|\to \infty$ with $y/x$ tending to some direction $b/a$, every term $Q,R,S,\dots$ of lower degree becomes negligible compared to $P$. So branches at infinity follow the directions $y/x=b/a$ where $P$ vanishes, and these are exactly the slopes of the real linear factors $ay-bx$ of $P$. Each real linear factor — counted with multiplicity — accounts for one or more pairs of opposite branches and dictates whether the asymptote is straight (rectilinear-asymptote-from-equation), parabolic (parabolic-asymptote), or higher-order curvilinear (curvilinear-asymptote).

Related pages

• chapter-7-on-the-investigation-of-branches-which-go-to-infinity
• rectilinear-asymptote-from-equation
• parabolic-asymptote
• curvilinear-asymptote
• classification-of-conics
• asymptotes-of-hyperbola
• order-of-an-algebraic-curve