# Category Archives: Algebraic Geometry

## Algebraic Geometry Jottings 18

Prev TOC Next

Up till now, I’ve been using power series to parametrize branches:

x(t) = a0+a1t+a2t2+ ⋯,      y(t) = b0+b1t+b2t2+ ⋯

If the branch passes through the origin, then a0=b0=0. In the last post, we established Facts 4 and 5, assuming that y(t)=t for all branches, so x(t) = a0+a1y+a2y2+ ⋯.

Filed under Algebraic Geometry

## Algebraic Geometry Jottings 17

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.

Filed under Algebraic Geometry

## Algebraic Geometry Jottings 16

The Resultant, Episode 5: Inside the Episode

The double-product form for the resultant:

$\text{res}_x(E,F) = a_m^n(y) b_n^m(y) \prod_{i=1}^m\prod_{j=1}^n (u_i-v_j)$  (1)

implies Fact 3:

Filed under Algebraic Geometry

## Algebraic Geometry Jottings 15

The Resultant, Episode 5 (The Finale)

Recap: The setting is an integral domain R, with fraction field K, and extension field L of K in which E(x) and F(x) split completely. E(x) and F(x) have coefficients in R. E(x) has degree m, F(x) degree n; we assume m,n>0. The main special case for us: R=k[y], K=k(y), so R[x]=k[x,y], and E and F are polynomials in x and y. As always, we assume k is algebraically closed.

Filed under Algebraic Geometry

## Algebraic Geometry Jottings 14

The Resultant, Episode 4

This episode has one sole purpose: to show that the two formulas for the resultant are equivalent. The next episode, the finale, will tie up some loose ends.

Filed under Algebraic Geometry

## Algebraic Geometry Jottings 13

The Resultant, Episode 3: Inside the Episode

So we have, at long last, several expressions for the resultant:

Filed under Algebraic Geometry

## Algebraic Geometry Jottings 12

The Resultant, Episode 3

Last time the linear operator

Φ: Kn[x]⊕Km[x] → Km+n[x]
Φ(p,q)=pE+qF

made its grand entrance, clothed in the Sylvester matrix. (Recall that Kn[x] is the vector space of all polynomials of degree <n with coefficients in K, likewise for Km[x] and Km+n[x].)

Filed under Algebraic Geometry

## Algebraic Geometry Jottings 11

The Resultant, Episode 2

By now you know the characters: the polynomials E(x) (degree m) and F(x) (degree n) with coefficients in an integral domain R, its fraction field K, and the extension field L of K in which E and F split completely:

Filed under Algebraic Geometry

## 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: