In 1856 Dirichlet made the following claim in a lecture:

# Monthly Archives: April 2020

## The Monoenergetic Heresy (Part 1)

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 curveover 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

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: *K _{n}*[

*x*]⊕

*K*[

_{m}*x*] →

*K*[

_{m+n}*x*]

Φ(

*p*,

*q*)=

*pE+qF*

made its grand entrance, clothed in the *Sylvester matrix*. (Recall that *K _{n}*[

*x*] is the vector space of all polynomials of degree <

*n*with coefficients in

*K*, likewise for

*K*[

_{m}*x*] and

*K*[

_{m+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

## Algebraic Geometry Jottings 10

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–u*_{1})···(*x–u _{m}*) and

*b*(

*x–v*

_{1})···(

*x–v*). (Repeated roots allowed.) We had our first formulas for the resultant:

_{n}Filed under Algebraic Geometry

## Algebraic Geometry Jottings 9

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

## Algebraic Geometry Jottings 8

Filed under Algebraic Geometry