# Monthly Archives: May 2020

## Algebraic Geometry Jottings 18

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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.