The Resultant, Episode 5: Inside the Episode

The double-product form for the resultant:

(1)

implies Fact 3:

The Resultant, Episode 5: Inside the Episode

The double-product form for the resultant:

(1)

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:

And now for something completely different.

Filed under Bagatelles, History

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…

The Resultant, Episode 3: Inside the Episode

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

Filed under Algebraic Geometry

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: *K _{n}*[

Φ(

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

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