At last we come to Kendig’s proof of Bézout’s Theorem. Although not long, it will take me a few posts to appreciate it in full.

Kendig starts by choosing favorable coordinates; among other desiderata, he wants to avoid intersections at infinity. From the standpoint of logical efficiency, this is the right call. But I’d rather go *through* these difficulties than *around* them, as much as possible, hoping to glean more insight.

Another complication: fractional power series (Puiseax series). Kendig introduces these in §3.3. I’ve been avoiding them, and I prefer to postpone the reckoning for one more post.

The argument, thus modified, has four steps.

- Show Facts 4 and 5 from post 15: the order of the resultant is the sum of the multiplicities on the x-axis (with some provisos), and the degree of the resultant is the sum of all multiplicities in the affine plane (same provisos).
- Adapt (1) to handle fractional power series.
- Homogenize
*E*and*F*and show how the homogenized resultant now counts the intersections at infinity as well. - Show that the homogenized resultant has degree
*mn*.

As usual, I’ll chew over the details.

Recall Kendig’s first definition of multiplicity, for a branch of *E* at an intersection with the curve *F* at the origin *O*. (For another intersection point *P*, move *P* to *O*.) First assume we’ve parametrized the branch via power series in a variable *t*, (*x _{E}*(

*t*),

*y*(

_{E}*t*)). Plug into the polynomial

*F*(

*x,y*) getting a power series

*F*(

*t*) =

*F*(

*x*(

_{E}*t*),

*y*(

_{E}*t*)). The order of

*F*(

*t*)—the degree of its lowest order term—is the multiplicity of the branch-curve intersection. Post 3 explored the intuition behind this: when we “perturb”

*E*or

*F*or both a little, an intersection of multiplicity

*r*typically splits into

*r*distinct intersections.

Remember also the local-global feature of Bézout’s Theorem: we need to total up all the branch-branch multiplicities at an intersection, and then sum this over all intersections. Kendig’s first definition already sums over the branches of the curve *F*. The second definition sums over horizontal lines when using res* _{x}*(

*y*), or vertical lines when using res

*(*

_{y}*x*). For example, the roses:

The green horizontal lines pass through the intersections; the green numbers at right are total multiplicities. As we’ve seen before, res_{x}(*y*)=16*y*^{14}(16*y*^{2}-5)^{2} has order 14, for the single intersection of multiplicity 14 on the x-axis. The top horizontal line has y-coordinate √5/4; we shift it down to be the new x-axis by substituting (*y*+√5/4) for *y* in the resultant. It would be painful to expand the result by hand, but we care only about are the lowest degree terms of 16(*y*+√5/4)^{14}, clearly a nonzero constant, and (16(*y*+√5/4)^{2}-5)^{2}, easily computed to be a constant times *y*^{2}. So we get total multiplicity 2 for this line. Likewise for the bottom horizontal line.

Recall the formulas for the resultant:

(1)

(2)

*E*(*x,y*) = *a _{m}*(

*y*)

*x*+···+

^{m}*a*

_{0}(

*y*) =

*a*(

_{m}*y*)(

*x–u*

_{1})···(

*x–u*) (3a)

_{m}*F*(

*x,y*) =

*b*)

_{n}(y*x*+···+

^{n}*b*

_{0}(

*y*) =

*b*)(

_{n}(y*x–v*

_{1})···(

*x–v*) (3b)

_{n}(I’ve made it explicit that the coefficients are polynomials in *y*.) Kendig proves Fact 4 (or rather a special case) using (1), but I prefer to use (2). The branches of *E* crossing the x-axis are essentially just the roots of the equation *E*(*x,y*) = 0; let’s assume these can be expressed as power series in *y* for *y* small enough. (This won’t work if a branch isn’t locally the graph of a function, but I’m postponing that issue.) Thus the *i*-th branch has the parametrization (*u _{i}*(

*y*),

*y*), where

*u*(

_{i}*y*) is a power series in

*y*. In other words, we have a parametrization (

*x*,

*y*) = (

*u*(

_{i}*t*),

*t*).

By Kendig’s first definition, the multiplicity of the intersection between the *i*-th branch of *E* and the curve *F* is just the order of *F*(*t*) = *F*(*u _{i}*(

*t*),

*t*) in

*t*. Since

*y*=

*t*, this is just the order of

*F*(

*u*(

_{i}*y*),

*y*) as a power series in

*y*. When you multiply power series, the orders add. Therefore, by eq.(2), the order of res

*(*

_{x}*y*) in

*y*is the sum of the multiplicities over all branches of

*E*crossing the x-axis, plus

*n*times the order of

*a*(

_{m}*y*). If

*a*(

_{m}*y*) is a constant, we get Fact 4 as stated.

How about the other horizontal lines? To get their y-coordinates, we factor the resultant, say

res* _{x}*(

*y*)

*= k*(

*y–c*

_{1})

^{r1}···(

*y–c*)

_{l}

^{rl}If, say, *c*_{1}=0, then *r*_{1} would be the order of res* _{x}*(

*y*), measuring the contribution from the x-axis. If

*c*

_{1}≠0, then we shift things up or down to

*make*it the x-axis, by substituting

*y*+

*c*

_{1}for

*y*. This works for any

*c*. We conclude that

_{i}*r*is the total multiplicity of the intersections on the line

_{i}*y*=

*c*

*(when*

_{i}*a*(

_{m}*y*) is constant). Since

*r*

_{1}+···+

*r*is the degree of res

_{l}*(*

_{x}*y*), we’re done!