Algebraic Geometry Jottings 16

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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:

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Algebraic Geometry Jottings 15

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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.

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Algebraic Geometry Jottings 14

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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.

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Weierstrass’s Smackdown of Dirichlet’s Principle

In 1856 Dirichlet made the following claim in a lecture:

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The Monoenergetic Heresy (Part 1)

The Emperor Heraclius.
Classical Numismatic Group, Inc. Wikimedia Commons

And now for something completely different.

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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…

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Algebraic Geometry Jottings 13

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The Resultant, Episode 3: Inside the Episode

So we have, at long last, several expressions for the resultant:

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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?

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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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