The Resultant, Episode 5: Inside the Episode
The double-product form for the resultant:
implies Fact 3:
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
Weierstrass’s Smackdown of Dirichlet’s Principle
In 1856 Dirichlet made the following claim in a lecture:
The Monoenergetic Heresy (Part 1)
The Emperor Heraclius.
Classical Numismatic Group, Inc. Wikimedia Commons
And now for something completely different.
Filed under Bagatelles, History
Escher’s Toroidal Print Gallery
If Art+Math brings one person to mind, it’s Escher. His tessellations present the best-known instance, but he did a lot more than that.
In April 2003, the mathematicians Bart de Smit and Hendrik Lenstra wrote a delightful article, Escher and the Droste effect, about Escher’s lithograph Prentententoonstelling. They pointed out that
We shall see that the lithograph can be viewed as drawn on a certain elliptic curve over the field of complex numbers…
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
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
