Last time we looked at Kendig‘s first definition of multiplicity. A branch of *E*, parametrized by (*x _{E}*(

*t*),

*y*(

_{E}*t*)), passes through the origin

*O*, as does the curve

*F*. Assume

*x*(

_{E}*t*) and

*y*(

_{E}*t*) are power series in

*t*. Plug them into the polynomial

*F*(

*x*,

*y*), getting a power series

*F*(

*t*). The order of

*F*(

*t*) (the degree of the first nonzero term) is the multiplicity of that intersection—that is, of the branch of

*E*with the entire curve

*F*at

*O*.

You’ll notice an asymmetry: we’ve defined the multiplicity for a *branch* of *E* with the *curve F*, at *O*. The elemental multiplicities, so to speak, are between two branches. For examples, these two branches of our running example, with *t* near 0:

*E*: *r* = –sin 3*t*

*F*: *r* = sin 2*t*

As we saw last time, to lowest order these branches are parametrized by and . Last time we plugged into *F*(*x*,*y*), and got something of order *t*^{6}. That checked out: the *E* branch had two double intersections with the “horizontal” branches of *F*, and two simple intersections with the “vertical” branches.

To limit our attention to just the branches, we eliminate *t* and write , . So we have a common parameter *x* for *E* and *F*, and (approximately) equations , with different constants *a* and *b*. The “level curve” idea says we should plug (*x*,*y*)=(*x*, *ax*^{2}) into *F*_{branch}(*x*,*y*)=*y*–*bx*^{2}, getting (*a–b*)*x*^{2}. Multiplicity 2, as expected.

More generally, say the branches (near *O*) of two curves *E* and *F* look like *y _{E}*=

*u*(

*x*),

*y*=

_{F}*v*(

*x*), for some power series

*u*(

*x*) and

*v*(

*x*). Plugging (

*x*,

*u*(

*x*)) into

*F*

_{branch}(

*x*,

*y*)=

*y*–

*v*(

*x*) gives

*u*(

*x*)–

*v*(

*x*). The lowest order of

*u*(

*x*)–

*v*(

*x*) is the multiplicity of the intersection. Geometrically this makes perfect sense: the branches are functions of

*x*near 0, and the (signed) vertical distance tells us the “degree of contact” between them.

Algebraically, we’ve taken two equations *y–**u*(*x*)=0 and *y*–*v*(*x*)=0 and eliminated *y*, getting *u*(*x*)–*v*(*x*)=0. This is a special case of the resultant. I’ll say more about it in a later post, but briefly: the resultant of *E*(*x*,*y*)=0 and *F*(*x*,*y*)=0 with respect to *y* is an equation in *x* alone, obtained by eliminating *y*. The two equations can have a common solution only for those values of *x* satisfying the resultant. We’ll write res* _{y}*(

*E*,

*F*)=0 for the resultant (with respect to

*y*). When

*E*(

*x*,

*y*) and

*F*(

*x*,

*y*) are polynomials in

*x*and

*y*, it turns out that res

*(*

_{y}*E*,

*F*) is a polynomial in

*x*. Likewise, res

*(*

_{x}*E*,

*F*) is the resultant with respect to

*x*, a polynomial in

*y*, telling you possible values of

*y*at an intersection of

*E*and

*F*.

Maybe you’re wondering, “Hey, *y–**u*(*x*) isn’t a polynomial in *x* and *y*!” Right you are. It’s a polynomial in *y* with coefficients in the ring of power series in *x*. And if *E*(*x*,*y*) is a polynomial in *x* and *y*, then we can treat it as a polynomial in *y* with coefficients in the ring of polynomials in *x* (or vice versa). So a common framework encompasses both cases.

Kendig’s second definition of multiplicity (§3.9) uses the resultant. Say *E* and *F* intersect at *O*, and suppose that *O* is their *only* intersection on the y-axis. One more proviso: treating *E* and *F* as polynomials in *y* with coefficients in the ring *k*[*x*], assume the leading coefficients are constants. (This rules out *xy*–1, for example.) The order of res* _{y}*(

*E*,

*F*)—the degree of the lowest order term—is the multiplicity at

*O*of the intersection of

*E*and

*F*. (See Kendig’s Theorem 3.3, p.62, for details.) This sums all the branch-branch multiplicities for the branches of

*E*and

*F*passing through

*O*. You can also use res

*(*

_{x}*E*,

*F*) provided

*O*is the only intersection on the x-axis.

If several intersections lie on the y-axis, then order of res* _{y}*(

*E*,

*F*) is the sum of all the multiplicities of all these intersections (with the same proviso about leading coefficients). Kendig’s proof, with a little tweaking, shows this.

For our running example,

res* _{x}*(

*E*,

*F*) = 16

*y*

^{14}(16

*y*

^{2}-5)

^{2}

and the multiplicity of *O* is 14, as we’ve learned. (I used SageMath to compute this.)

Returning to the general case: at the most granular level, we have the branch-branch multiplicities. Summing all these at a point *P* gives the intersection multiplicity at *P*. Fulton denotes this , and calls it the *intersection number*. (We’ve been letting *P*=*O*, but of course you can always translate *P* to the origin.) For Bézout’s theorem, we have to sum these over all the intersections of *E* and *F*.

Before turning to Fulton’s definitions in the next post, a remark. I’ve been assuming that any branch of *E*(*x*,*y*)=0 near *O* has a parametrization (*x _{E}*(

*t*),

*y*(

_{E}*t*)), with

*x*(

_{E}*t*) and

*y*(

_{E}*t*) power series in

*t*. Actually more is true: you can always take

*x*(

_{E}*t*) to be a power of

*t*. (See Kendig §3.4.) However, the special form (

*x*,

*u*(

*x*)) that we used above, with

*u*(

*x*) a power series in

*x*, is not always possible.