Better video (and explanation!) of this week's mathematical conspiracy; a thread. 1/npic.twitter.com/n2fdNyvbWU
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The space of quadratic polynomials can be identified with R^3; we think of the function ax^2+bx+c as the point (a,b,c). The polynomials with integer coefficients form the cubical lattice. 2/npic.twitter.com/jybRhcFgjr
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The discriminant of ax^2+bx+c, namely b^2-4ac determines whether the polynomial has two real roots (positive discriminant) or complex conjugate complex roots (negative discriminant). The discriminant=0 surface is a cone. 3/npic.twitter.com/slQicyTOi3
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Thus; the space of quadratics looks like Minkowski space, with the discriminant as the norm. Standing at the origin of this space and looking "straight up inside the cone" we see the polynomials with complex conjugate roots. 4/npic.twitter.com/JinvV4DOMR
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We get a better view of this perspective through projectivization (which is a useful thing to do if we are interested in studying polynomials through their roots anyway, as f(x) and cf(x) have the same roots). Here I've colored and the polynomials by their discriminant. 5/npic.twitter.com/Mrs3jXBVXc
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That's the picture: green polynomials have complex roots, blue have two real roots. Where does the motion come from? Any homeo A:C->C of the plane induces a transformation of (projectivized) polynomials taking the poly f with roots z,w to the poly A.f with roots A(z), A(w). 6/n
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Let f_t be the horizontal translations of C, given by z-> z+t. This induces a shearing motion on the space of quadratics. 7/npic.twitter.com/E31EtW4z8U
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I would be interested in figuring out how this shearing motion affects lattice sums over Z^3.
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