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?
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
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
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
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
Filed under Algebraic Geometry
Fulton characterizes the intersection number, I(P, E∩F), with seven properties. Let me just repeat the last one:
Filed under Algebraic Geometry
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
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