The Resultant, Episode 3: Inside the Episode
So we have, at long last, several expressions for the resultant:
Category Archives: Math
Wallpaper Groups
Escher: Alhambra Sketch
I first learned as a kid that “there are only 17 basically different wallpapers” from W.W.Sawyer’s Prelude to Mathematics. (The quote appears on p.102. Aside: this remains an excellent gift for a youngster with a yen for math.) I remember my father pointing out the absurdity of this claim: are all mural wallpapers of van Gogh’s paintings basically the same?
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:
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
Filed under Algebraic Geometry
Algebraic Geometry Jottings 7
Fulton characterizes the intersection number, I(P, E∩F), with seven properties. Let me just repeat the last one:
- 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(E∩F) with seven properties (§3.3). (Fulton calls it the intersection number. Also, he writes I(P,E∩F) 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
