Author Archives: Michael Weiss

Algebraic Geometry Jottings 12

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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].)

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Algebraic Geometry Jottings 11

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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:

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Algebraic Geometry Jottings 10

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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:

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Algebraic Geometry Jottings 9

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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!

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Algebraic Geometry Jottings 8

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This will be a “chewing” post.

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Algebraic Geometry Jottings 7

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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.

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Algebraic Geometry Jottings 6

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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,

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Algebraic Geometry Jottings 5

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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:

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Algebraic Geometry Jottings 4

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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.

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Algebraic Geometry Jottings 3

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

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