Patrick Morton

Bifurcations and discriminants for polynomial maps

We characterize the bifurcations of polynomial maps algebraically, by means of the discriminants of the polynomials whose roots are the periodic orbits. These discriminants are computed explicitly, in terms of multiplier polynomials. This approach affords a generalization of the notion of bifurcation to the broader context of iteration of polynomials over an integral domain.

Quasi-Symmetric 3-Designs and Elliptic Curves

A quasi-symmetric t-design is a t-design with two block intersection sizes p and q (where

On the elgenvectors of Schur's matrix

p < q

Two-weight ternary codes and the equation y2 = 4 × 3a + 13

). Quasi-symmetric 3-designs are classified with

The integer points on three related elliptic curves

p = 1

Pattern spectra, substring enumeration, and automatic sequences

. The only nontrivial examples are the 4-(23, 7, 1) Witt design, and its residual, a 3-(22, 7, 4) design. This proves a conjecture of Sane and Shrikhande. The method is to reduce the classification problem to that of finding all integer points on the elliptic curves

A new characterization of the integer 5906

y^2 = x^3 - 11x^2 + 32x

The quartic Fermat equation in Hilbert class fields of imaginary quadratic fields

and

Solutions of the cubic Fermat equation in ring class fields of imaginary quadratic fields (as periodic points of a 3-adic algebraic function)

y^2 = x^3 - 4x + 4

A correction to Hasse's version of the Grunwald–Hasse–Wang theorem

.