general-equation-multiple-point

General Equation of a Curve with a Multiple Point

Summary: Synthetic counterpart to singular-points-by-jet. A curve passes through $M=(p,q)$ iff its equation has the form $P(x-p)+Q(y-q)=0$ with $P,Q$ polynomials in $x,y$ (and the linear factors $x-p$, $y-q$ irreducible against them). It has a $k$-fold point at $M$ iff the equation has the form $P_0(x-p{)}^k+P_1(x-p{)}^{k-1}(y-q)+\cdots +P_k(y-q{)}^k=0$, i.e., is homogeneous of degree $k$ in the translated coordinates with polynomial coefficients. From these forms, order constraints follow via the chapter-4 line-intersection bound: cubics have at most one double point and no triple points; quartics have at most one triple point; fifth order is the minimum for a quadruple point; eighth order is the minimum for two quadruple points. Euler’s stated bound for double points on order-$n$ curves is $r\le n-2$, an under-count for $n\ge 4$ relative to the modern Plücker bound $r\le \left(\genfrac{}{}{0ex}{}{n-1}{2}\right)$.

Sources: chapter13 (§§299–302).

Last updated: 2026-05-07.

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A curve through a given point (§299)

Let $M=(p,q)$ and let $P,Q,R,S,\dots$ be polynomials in $x,y$. The equation $P(x-p)+Q(y-q)=0$ expresses a curve passing through $M$, provided $P$ is not divisible by $y-q$ and $Q$ is not divisible by $x-p$ (otherwise the factors $x-p$ or $y-q$ that make the curve pass through $M$ could be eliminated by division). Conversely, every curve passing through $M$ has an equation of this form. The point $M$ is a simple (non-singular) point unless the equation can be brought to one of the multiple-point forms below.

Double point (§300)

The curve has a double point at $M$ iff its equation has the form $P(x-p{)}^2+Q(x-p)(y-q)+R(y-q{)}^2=0,$ provided this form is not lost through division. The three monomials are exactly the homogeneous quadratic monomials in the translated coordinates, so this equation exhibits the second jet at $M$.

Order consequences

• Second-order line cannot have a double point. If the curve is of order 2, then $P,Q,R$ must be constants. But then the equation is a quadratic form in $(x-p)$ and $(y-q)$ — a product of two straight lines through $M$, not a true second-order curve. So a conic has no double point.
• Third-order line has at most one double point. If a cubic had two double points, the straight line through them would meet the curve in $2+2=4$ points, contradicting the cubic-line bound of 3.
• Fourth-order line has at most two double points (Euler’s stated claim).
• Fifth-order line has at most three double points (Euler’s stated claim).
• In general, $r$ doubles on an order-$n$ curve satisfy $r\le n-2$ (Euler’s stated claim).

A flag on Euler’s general bound

The line-test argument Euler gives only refutes configurations where the doubles lie on a common line: $r$ collinear doubles force $2r$ intersections, so $2r\le n$, i.e., $r\le n/2$ collinear doubles. This is consistent with everything for $r$ collinear doubles, but does not refute non-collinear configurations. For $n=4$, three non-collinear doubles ARE possible — a quartic with three nodes is a rational quartic. The classical bound (proven later, via Plücker / arithmetic genus) is

$r\le \left(\genfrac{}{}{0ex}{}{n-1}{2}\right)=\frac{(n-1)(n-2)}{2},$

which for $n=4$ gives $r\le 3$, not Euler’s $r\le 2$. Euler’s $n-2$ bound is correct for $n=3$ (both formulas give 1) but undercounts for $n\ge 4$. Flag for verification: the source-text wording is explicit, “Likewise a fourth order line has no more than two double points.”

Triple point (§301)

If $M$ is a triple point, the equation has the form $P(x-p{)}^3+Q(x-p{)}^2(y-q)+R(x-p)(y-q{)}^2+S(y-q{)}^3=0,$ homogeneous of degree 3 in the translated coordinates with polynomial coefficients $P,Q,R,S$ in $x,y$.

Order consequences

• Third-order line has no triple point. If $P,Q,R,S$ were constants, the equation would factor (over $\mathbb{C}$) into three linear factors $\alpha (x-p)+\beta (y-q)$, giving three concurrent straight lines — not a true cubic curve.
• Fourth-order line has at most one triple point. Two would force a line through them to meet the quartic in $3+3=6$ points, contradicting the quartic-line bound of 4.
• Fifth-order line has at most one triple point. Two would force 6 line-intersections, contradicting bound 5.
• Sixth-order line can have two triple points. Two triples force exactly 6 intersections, hitting the quartic-line bound of 6 tightly.

For triple points the line-test argument is sharper than for doubles, because each triple contributes 3 to the line count.

Quadruple point (§302)

If $M$ is a quadruple point, the equation has the form $P(x-p{)}^4+Q(x-p{)}^3(y-q)+R(x-p{)}^2(y-q{)}^2+S(x-p)(y-q{)}^3+T(y-q{)}^4=0.$

Order consequences

• Fifth order is the minimum for a quadruple point (lower orders would force the equation to factor into linear pieces).
• Two quadruple points need order ≥ 8 (each contributes 4 line-intersections; two contribute 8; the line-intersection bound forces $n\ge 8$).
• The same recipe extends to quintuple and higher multiplicities.

Reading the local picture from the form (§302)

Once the equation is in multiple-point form $\sum P_i(x-p{)}^{k-i}(y-q{)}^i=0$, substitute $x=p+t$, $y=q+u$ (with $t,u$ replacing $x-p,y-q$), and the resulting equation in $(t,u)$ supplies the lowest non-vanishing homogeneous form near $M$. From this, the tangent directions of the branches and the presence of conjugate ovals can be read off directly. This is the round trip: starting from the synthetic form (multiple-point-prescribing equation) and recovering the analytic picture (jet-based tangent directions and branch counts).

The master constraint

Each $k$-fold point at $M$ contributes $k$ to the count of intersections of the curve with any straight line through $M$. Since [[line-curve-intersection-bound|a line meets an order-$n$ curve in at most $n$ points]], multiple points $M_1,M_2,\dots ,M_r$ of multiplicities $k_1,\dots ,k_r$ that lie on a common straight line must satisfy

$k_1+k_2+\cdots +k_r\le n.$

This is the master constraint behind every per-order ceiling above. Euler stops at this linear-test version; the sharper quadratic and higher tests (using a curve of degree $>1$ as test object) lead to the modern Plücker formulas, which give tighter bounds on configurations of singularities not constrained to a line.

Related pages

• chapter-13-on-the-dispositions-of-curves — chapter summary.
• singular-points-by-jet — analytic counterpart: characterizing multiple points by their lowest non-vanishing form.
• line-curve-intersection-bound — the master constraint behind every order ceiling here.
• multiple-points-on-curves — chapter 12’s discriminant-collision treatment of the same multiple-point phenomena.
• determinations-of-a-general-equation — counts the free parameters that allow these multiple-point conditions to be imposed.