The Truth about Truth
MW: A little while back, I noted something delicious about the history of mathematical logic:
Filed under Conversations, Peano Arithmetic
Up till now, I’ve been using power series to parametrize branches:
x(t) = a0+a1t+a2t2+ ⋯, y(t) = b0+b1t+b2t2+ ⋯
If the branch passes through the origin, then a0=b0=0. In the last post, we established Facts 4 and 5, assuming that y(t)=t for all branches, so x(t) = a0+a1y+a2y2+ ⋯.
Filed under Algebraic Geometry
At last we come to Kendig’s proof of Bézout’s Theorem. Although not long, it will take me a few posts to appreciate it in full.
Filed under Algebraic Geometry
The Resultant, Episode 5: Inside the Episode
The double-product form for the resultant:
implies Fact 3:
Filed under Algebraic Geometry
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
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
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.
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…
The Resultant, Episode 3: Inside the Episode
So we have, at long last, several expressions for the resultant:
Filed under Algebraic Geometry
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?
Diagonal Argument 