## Algebraic Geometry Jottings 10

The Resultant, Episode 1: Inside the Episode

In Episode 1 of our miniseries, “The Resultant”, the characters were introduced: integral domain R with fraction field K and extension field L, and polynomials E(x) and F(x) in R[x], factoring completely in L as a(x–u1)···(x–um) and b(x–v1)···(x–vn). (Repeated roots allowed.) We had our first formulas for the resultant:

Filed under Algebraic Geometry

## Algebraic Geometry Jottings 9

The Resultant, Episode 1

Time to discuss the resultant; we’ll need it for Kendig’s proof of Bézout’s theorem, but it has other uses too. The story will take several episodes, plus extras. Like a miniseries!

Filed under Algebraic Geometry

## Algebraic Geometry Jottings 8

This will be a “chewing” post.

Filed under Algebraic Geometry

## Algebraic Geometry Jottings 7

Fulton characterizes the intersection number, I(P, EF), with seven properties. Let me just repeat the last one:

1.  I(E∩F) = I(E∩(F+AE)) for any A.

Filed under Algebraic Geometry

## Algebraic Geometry Jottings 6

The tome Commutative Algebra by Zariski and Samuel opens with the memorable sentence, “This book is the child of an unborn parent.” As Zariski explains,

Filed under Algebraic Geometry

## Algebraic Geometry Jottings 5

We’ve been looking at Kendig‘s two definitions of intersection multiplicity; now let’s look at Fulton‘s.

Fulton characterizes the multiplicity I(EF) with seven properties (§3.3). (Fulton calls it the intersection number. Also, he writes I(P,EF) for the intersection number at P. I’ll usually assume P is the origin O, and omit writing it.) The last three properties stand out:

Filed under Algebraic Geometry

## Algebraic Geometry Jottings 4

Last time we looked at Kendig‘s first definition of multiplicity. A branch of E, parametrized by (xE(t),yE(t)), passes through the origin O, as does the curve F. Assume xE(t) and yE(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.

Filed under Algebraic Geometry

## Algebraic Geometry Jottings 3

Bézout’s theorem requires us to count intersection points according to their multiplicity. OK, what’s multiplicity? (Fulton uses the phrase intersection number.)

Filed under Algebraic Geometry

## Algebraic Geometry Jottings 2

As I said last time, I’m learning some algebraic geometry, starting with Bézout’s theorem, and using Fulton’s Algebraic Curves and Kendig’s A Guide to Plane Algebraic Curves as the texts. Right now we’re looking at this example from Fulton: