The Resultant, Episode 5: Inside the Episode
The double-product form for the resultant:
implies Fact 3:
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.
In 1856 Dirichlet made the following claim in a lecture:
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.
We shall see that the lithograph can be viewed as drawn on a certain elliptic curve over the field of complex numbers…
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?
The Resultant, Episode 3
Last time the linear operator
Φ: Kn[x]⊕Km[x] → Km+n[x]
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].)