Intersection Product Degree Bound

Summary: Two algebraic curves of orders $m$ and $n$ meet in at most $mn$ points (§§496–498). Euler establishes the bound case by case — two conics ($\le 4$), conic and cubic ($\le 6$), two cubics ($\le 9$), two quartics ($\le 16$) — both by algebraic elimination and by the geometric argument that a curve of order $n$ decomposes (as a complex curve) into $n$ straight lines, each of which meets the other curve in at most $m$ points. The bound governs how to factor a target equation degree $N$ for a construction: pick $m\cdot n=N$ if possible, otherwise a slightly larger product with simpler factors.

Sources: chapter20 (§§496–498).

Last updated: 2026-05-12.

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The general statement (§498)

Two curves of orders $m$ and $n$ have at most $mn$ intersection points.

Euler’s $mn$ bound is the real, affine, multiplicity-free shadow of what becomes Bézout’s theorem in projective complex geometry: there the count is exactly $mn$, including multiplicities and points at infinity over $\mathbb{C}$. The chapter-19 phenomenon of complex-intersections is the difference between Euler’s bound and Bézout’s equality.

Two derivations

Algebraic, via elimination

For two equations of $y$-degrees $m$ and $n$ with all-degree-$\le d$ in $z$, the eliminant from §§474–482 is a polynomial in $z$ of degree at most $mn$:

• Two conics ($m=n=2$). Write each as $P+Qy+R{y}^2=0$ and $p+qy+r{y}^2=0$ with $P,p$ quadratic in $z$, $Q,q$ linear, $R,r$ constant. The $(2,2)$ eliminant of §479 has $z$-degree at most 4. So two conics meet in at most 4 points (§496).

• Conic and cubic ($m=2,n=3$). Write one as $P+Qy+R{y}^2=0$ and the other as $p+qy+r{y}^2+s{y}^3=0$ with $P$ quadratic, $Q$ linear, $R$ constant, $p$ cubic, $q$ quadratic, $r$ linear, $s$ constant. The §480 eliminant is sixth degree (§497).

• Two cubics ($m=n=3$). The §481 eliminant has degree 9.

• Two quartics ($m=n=4$). The §482 cascade gives degree 16.

Geometric, via complex-curve decomposition

A second-order curve can be a complex curve — two straight lines (figure 16, §63). A straight line meets a curve of order $m$ in at most $m$ points (line-curve-intersection-bound); two lines, taken together as a degenerate “second-order curve,” meet a curve of order $m$ in at most $2m$ points. Hence two conics (one of them possibly the degenerate two-line curve, by continuity argument) meet in at most $2\cdot 2=4$ points (§496).

Same argument for higher orders: a degenerate order-$n$ curve is a union of $n$ lines, each meeting an order-$m$ curve in at most $m$ points, so the total is $\le mn$. By continuity, the generic case is also bounded by $mn$.

Cumulative table

$(m,n)$	Bound	Eliminant in §§	Notes
$(1,n)$	$n$	line substitution	line-curve-intersection-bound (chapter 4)
$(2,2)$	$4$	§479	two conics — exactly the quartic case
$(2,3)$	$6$	§480	conic ∩ cubic
$(3,3)$	$9$	§481	two cubics
$(4,4)$	$16$	§482	two quartics
$(m,n)$	$mn$	by induction	general

Choosing factors for a target degree

To construct an equation of degree $N$, choose $m,n$ with $mn=N$ — or, if $N$ is prime or has only awkward factorizations, $mn\ge N$ and discard the excess intersections:

• $N=100$ factors as $10\cdot 10$ or $5\cdot 20$ or $4\cdot 25$, etc. Two curves of order 10 each is one choice; one of order 5 plus one of order 20 is another.
• $N=39=3\cdot 13$ — the factor 13 is awkward (cubic-genera enumeration in chapter 9, quartic genera in chapter 11 — order-13 curves are unenumerated and visually intractable). Euler suggests instead choosing $m,n$ with $mn=42=6\cdot 7$: two curves of orders 6 and 7 give 42 intersections, three of which are spurious roots ($mn-N=3$) that can be discarded.
• $N$ prime. Choose $m\cdot n=N+k$ for the smallest $k$ giving a convenient factorization.

The general principle: lower-order curves are visually simpler and the genera (cubic species in chapter 9, quartic genera in chapter 11) are catalogued. So $6\cdot 7=42$ is a simpler construction than $1\cdot 13$ for $N=13$, even though the latter is exact: an order-13 curve is harder to draw than an order-7 curve.

Connection to chapter 19

Chapter 19 set up the elimination tables without explicitly stating the $mn$ bound: it focused on the procedure and on complex intersections. Chapter 20 names the bound, gives the geometric proof via complex-curve degeneration, and packages the result for use in construction problems.

Figures

Figures 15–18

Related pages

• construction-of-equations
• chapter-20-on-the-construction-of-equations
• line-curve-intersection-bound
• elimination-of-ordinate
• complex-curves
• complex-intersections
• intersection-of-two-curves