Bezout's Theorem - wiki

More documents

Recommendations

Info

${\infty}$ preparation and division Theorems - wiki - University of ...$

1/11 Say, P := (P1, · · · , Pn): C n → C n , assume #P −1 (0) < ∞ Then (introductory lectures, October 2009) ⇒ ∞ > dimC C[x] = (P) · C[x] a∈P−1 dimC (0) C[x]a (P) · C[x]a deg Pj Let LPj(x) := HPj(0, x), where HPj(x0, x) := x0 · Pj( x x0 ). Assume also {LP1(x) = · · · = LPn(x) = 0} = {0} (∗) (so called “no solutions at ∞”) for C n ↩→ P n ⇒
Bezout’s <strong>Theorem</strong> Proof of (*) ⇒ a. 2/11 a. P : C n → C n is proper finite to one b. deg C a P = deg P = deg Pj a∈P −1 (0) 1≤j≤n Say, ξk ∈ C n , assume ξk ↗ ∞ and all |Pj(ξk)| ≤ C < ∞ ak := ξk ξk ∈ S 1 ⇒ ∃ subsequence ak → a = 0 ρk := 1 ξk → 0, 0 < dj = deg Pj
Page 1: Bezout’s Theorem Jooho Jeremie Ki
Page 5 and 6: 4/11 y is a regular value for LP
Page 7 and 8: 6/11 Step 3 is deg C 0 LP = j deg
Page 9 and 10: 8/11 Recall. deg LPj = deg Pj = dj,
Page 11 and 12: 10/11 Step 1. ∀a ∈ {Jφ|U = 0}

Bezout's Theorem - wiki

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?