Indeterminate-Multiplier Elimination

Summary: An alternative to the cascaded subtractions of elimination-of-ordinate. To eliminate $y$ from two polynomial equations of degrees $m$ and $n$ in $y$, multiply each by a polynomial in $y$ with indeterminate coefficients $A,B,C,\dots$ and $a,b,c,\dots$, then match coefficients of every power of $y$ in the two products. This yields a system of $m+n-1$ linear equations in $m+n-1$ unknowns whose solution provides the eliminant. Euler’s §§483–485 — essentially a Bézout/Sylvester-style resultant construction.

Sources: chapter19 (§§483–485).

Last updated: 2026-05-12.

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Setup (§483)

Given

$\text{I: }P{y}^m+Q{y}^{m-1}+R{y}^{m-2}+S{y}^{m-3}+\cdots =0,$ $\text{II: }p{y}^n+q{y}^{n-1}+r{y}^{n-2}+s{y}^{n-3}+\cdots =0,$

with $P,Q,R,\dots ,p,q,r,\dots$ polynomials in $z$ alone, we want a single equation in $z$ alone that the abscissas of intersections must satisfy. The strategy: form products

${\Pi }_1=\text{I}\cdot M(y),{\Pi }_2=\text{II}\cdot N(y),$

with $M(y)$ and $N(y)$ polynomial in $y$ with unknown (indeterminate) coefficients, chosen so that the difference ${\Pi }_1-{\Pi }_2$ has every power of $y$ cancel — except possibly the constant term in $y$, which is the desired equation in $z$.

Counting the indeterminates (§483)

Choose

$M(y)=p{y}^{k-n}+A{y}^{k-n-1}+B{y}^{k-n-2}+C{y}^{k-n-3}+\cdots ,$ $N(y)=P{y}^{k-m}+a{y}^{k-m-1}+b{y}^{k-m-2}+c{y}^{k-m-3}+\cdots ,$

so that the leading $y^k$ terms in ${\Pi }_1$ and ${\Pi }_2$ match automatically ($Pp=Pp$). The number of “free” indeterminate letters in $M$ is $k-n$ (the $A,B,C,\dots$), in $N$ is $k-m$ (the $a,b,c,\dots$). Total free letters: $2k-m-n$.

The matching condition cancels coefficients of $y^{k-1},y^{k-2},\dots ,y^1$, which is $k-1$ equations. Setting number of unknowns = number of equations requires

$2k-m-n=k-1⟹\boxed{k=m+n-1.}$

Then the $k-1=m+n-2$ matching equations exactly determine the $2k-m-n=m+n-2$ indeterminates, leaving the constant-in-$y$ coefficient as the eliminant.

The matching equations (§484)

Writing out coefficient comparisons of $y^{k-1},y^{k-2},y^{k-3},\dots$ in ${\Pi }_1$ and ${\Pi }_2$:

\text{$y^k$:} \quad & Pp = Pp \quad \text{(identity)} \\ \text{$y^{k-1}$:} \quad & Pa + Qp = pA + qP \\ \text{$y^{k-2}$:} \quad & Pb + Qa + Rp = pB + qA + rP \\ \text{$y^{k-3}$:} \quad & Pc + Qb + Ra + Sp = pC + qB + rA + sP \\ & \vdots \end{aligned}$$ These are $m + n - 1$ linear equations in the $m + n - 2$ unknowns $A, B, C, \ldots, a, b, c, \ldots$ (counting the identity at the top). The first equation is an identity; the remaining $m + n - 2$ pin down all the indeterminates. The *final* coefficient — the one with no $y$ at all — does not contain any unknowns and is the desired equation in $z$. ## Indeterminate multipliers $\alpha, \beta, \gamma$ (§485) Solving the matching equations directly is awkward. Euler introduces *further* indeterminates $\alpha, \beta, \gamma, \ldots$ to streamline the substitutions. He demonstrates on the (2, 3) case: $$\text{I: } Py^2 + Qy + R = 0, \qquad \text{II: } py^3 + qy^2 + ry + s = 0.$$ Here $m = 2$, $n = 3$, so $k = 4$. Multiply I by $py^2 + ay + b$ (free letters $a, b$, count $= k - n = 1$ — but Euler uses two for symmetry of method), and II by $Py + A$ (free letter $A$). The matching equations are $$\begin{aligned} Pp &= Pp, \\ Pa + Qp &= pA + qP = \alpha, \\ Pb + Qa + Rp &= qA + rP = \beta, \\ Qb + Ra &= rA + sP, \\ Rb &= sA. \end{aligned}$$ The first row is an identity; the last is the final equation in $z$ (the *eliminant*) once the indeterminates have been resolved. The intermediate rows provide solving equations. Resolving step by step: - Row 2: $a = (\alpha - Qp)/P$ and $A = (\alpha - qP)/p$. - Row 3: $b = \beta/P - Qa/P - Rp/P$, leading after substitution to $$b = \frac{\alpha(Pq - Qp)}{P^2 p} + \frac{Q^2 p^2 - P^2 q^2}{P^2 p} + \frac{Pr - Rp}{P}.$$ - Substituting into row 4 yields, after multiplying through by $P^2 p$, $$\alpha Q(Pq - Qp) + \alpha P(Rp - Pr) - Q(Pq - Qp)(Qp + Pq) + PQp(Pr - 2Rp) + P^3 qr - P^3 ps = 0,$$ which gives $$\alpha = \frac{P^2 Qq^2 - Q^3 Qpr - P^2 Qpr + 2PQRp^2 - P^3 qr + P^3 ps}{PQq - Q^2 p + PRp - P^2 r}.$$ (Euler's printed expression in §485 has a slightly different but equivalent shape.) - Substituting into row 5 yields a *second* expression for $\alpha$: $$\alpha = \frac{P^2 Rq^2 - Q^2 RP^2 - P^2 Rpr + PR^2 p^2 - P^3 qs}{PRq - QRp - P^2 s}.$$ - Equating the two expressions for $\alpha$ produces the desired equation in $z$ alone — and Euler notes (§485 closing) that it has the same form as the (2, 3) eliminant of §480. ## Why this method (§483 framing) > Although this method for eliminating one unknown from two equations has wide applications, still we will add another method which does not require so many repeated substitutions. (source: chapter19, §483) The cascaded-subtraction method of [[elimination-of-ordinate]] requires $\min(m, n)$ passes of doubling complexity. The indeterminate-multiplier method instead solves a linear system *all at once*, which scales better for higher $m, n$ and reveals the symmetric structure of the eliminant. It is essentially the **resultant** of two polynomials in $y$, viewed as a determinantal condition for them to share a root — modern formulation: $\operatorname{Res}_y(\text{I}, \text{II})$ as the determinant of the Sylvester matrix. Euler does not phrase the construction matrix-theoretically; he leaves it as a system to be solved by hand. But the framework of multiplying by indeterminate polynomials, then matching coefficients, is exactly the Bézout setup that Cayley and Sylvester would formalize a century later. ## Relation to Bézout's theorem The total degree of the eliminant — the polynomial that the abscissas of intersections satisfy — is at most $mn$ when curves of orders $m$ and $n$ in the variables $(z, y)$ are eliminated for $y$. Counting actual intersections (real + complex, in projective space, with multiplicity) gives exactly $mn$ — the modern Bézout theorem. Euler hints at this in §457 but the systematic $mn$ count is developed in the next chapter. ## Related pages - [[elimination-of-ordinate]] - [[intersection-of-two-curves]] - [[complex-intersections]] - [[chapter-19-on-the-intersection-of-curves]]